text:
\section{Introduction}
Fractional calculus, which is a generalization of integer order differentiation and $n-$fold integration, has been successfully applied in different branches of science and engineering. Differential equations with fractional-order derivatives (or integrals) are generally called fractional differential (or integral) equations. In recent past, fractional order differential equations have been used in several biological systems to explore the underlying dynamics  \cite{CuiYang14, DasGupta09, DasGupta11}. Here we assume an ecological system where a prey population grows logistically and a predator population feeds on this prey population. Now assume that the prey population is infected by some microparasites. In presence of infection, our prey population is divided into two subpopulations, viz. susceptible prey and infected prey. Since infected preys are weaken and cannot easily escape predation, predators disproportionately consume large number of infected prey \cite{LM96}. In such case, if there are sufficient numbers of infected prey, it may be assumed that the growth rate of predator is maintained mainly by consuming infected prey. If it is also considered that healthy preys can only give birth, infection transmits horizontally and predation process follows type II response function then we have the following eco-epidemiological system:
\begin{eqnarray}\label{Eco-epidemiological integer order model}
\frac{dS}{dt} & = & rS\bigg(1-\frac{S+I}{K}\bigg) - \lambda IS,\nonumber \\
\frac{dI}{dt}& = & \lambda IS-\frac{mIP}{a+I} - \mu I, \\
\frac{dP}{dt}& = & \frac{\theta IP}{a+I}-dP. \nonumber
\end{eqnarray}
The state variables $S(t), I(t)$ and $P(t)$ represent, respectively, the densities of susceptible, infected and predator populations at time $t$. Here $r$ is intrinsic birth rate of prey, $K$ is the environmental carrying capacity, $\lambda$ is the force of infection, $m$ is the maximum prey attack rate, $\mu$ is the death rate of infected prey, $\theta$ ($0<\theta\leq1$) is the conversion efficiency, $a$ is the half saturation constant and $d$ is the death rate of predator, All parameters are assumed to be positive from biological point of view. Readers are referred to \cite{ChattoBairagi01} for more discussion about the model. Note that it is an integer order system of differential equations and its dynamics was studied by Chattopadhyay and Bairagi \cite{ChattoBairagi01}.

Considering the fractional derivatives in the sense of Caputo derivative, and assuming $0<\alpha\leq1$, we have the following fractional order eco-epidemiological model corresponding to the model (\ref{Eco-epidemiological integer order model}):
\begin{eqnarray}\label{Eco-epidemiological fractional order model}
^{c}_{0} D^{\alpha}_{t}S & = & rS\bigg(1-\frac{S+I}{K}\bigg) - \lambda IS, \nonumber \\
^{c}_{0} D^{\alpha}_{t}I & = & \lambda IS-\frac{mIP}{a+I} - \mu I, \\
^{c}_{0} D^{\alpha}_{t}P & = & \frac{\theta IP}{a+I}-dP, \nonumber
\end{eqnarray}
where $^{c}_{0} D^{\alpha}_{t}$ is the Caputo fractional derivative. The main advantage of Caputo's approach is that the initial conditions for the fractional differential equations with Caputo derivatives takes the similar form as for integer-order differential equations \cite{CuiYang14, Podlubny99}, i.e., it has advantage of defining integer order initial conditions for fractional order differential equations. We analyze system (\ref{Eco-epidemiological fractional order model}) with the initial conditions $$S(0)>0, I(0)>0, P(0)>0.$$

The paper is organized as follows. In Section $2$, we give some useful theorems and lemmas in relation to fractional order differential equations. Well-posedness and dynamical behavior of the model are presented in Sections $3$ and $4$, respectively. Extensive numerical computations are presented in Section $5$ and the paper ends with a summary in Section $6$.
\section{Important Results}

\noindent\textbf{Theorem 1.} \cite{Matignon96} The following autonomous system
\begin{equation}\nonumber
^{c}_{t_{0}} D^{\alpha}_{t} x(t) = Ax, ~x(0)= x_{0},
\end{equation}
with $0<\alpha\leq1, x\in \Re^{n}$ and $A\in \Re^{n\times n}$ is asymptotically stable if and only if $\mid arg(\lambda)\mid >\frac{\alpha \pi}{2}$ is satisfied for all eigenvalues of the matrix $A$. Also, this system is stable if and only if $\mid arg(\lambda)\mid \geq \frac{\alpha \pi}{2}$ for all eigenvalues of the matrix $A$ with those critical eigenvalues satisfying $\mid arg(\lambda)\mid =\frac{\alpha \pi}{2}$ having geometric multiplicity of one. The geometric multiplicity of an eigenvalue $\lambda$ of the matrix $A$ is the dimension of the subspace of vectors $v$ for which $Av = \lambda v$.\\

\noindent\textbf{Theorem 2.} \cite{Petras11} Consider the following commensurated fractional order system
\begin{equation}\label{Stability condition} \nonumber
^{c}_{t_{0}} D^{\alpha}_{t} x(t) = f(x), x(0)= x_{0}
\end{equation}
with $0<\alpha\leq1, x\in \Re^{n}$ and $f: \Re^{n}\rightarrow \Re^{n}$ i.e., $f=(f_1, f_2 ----f_n)^T$. The equilibrium points of the  above system are calculated by solving the equation $f(x) = 0$. These equilibrium points are locally asymptotically stable if all eigenvalues $\lambda_{i}$ of the jacobian matrix $J = \frac{\partial f}{\partial x}$ evaluated at the equilibrium points satisfy $\mid arg(\lambda_{i})\mid >\frac{\alpha \pi}{2}$.\\

\noindent\textbf{Lemma 1} \cite{OdibatShawagfeh07}~~~(Generalized Mean Value Theorem) Suppose that$f(t)\in C[a,b]$ and $D^{\alpha}_{a}f(t)\in C(a,b]$ with $0<\alpha\leq1$, then we have
\begin{equation}\nonumber
f(t) = f(a) + \frac{1}{\Gamma(\alpha)}(D^{\alpha}_{a}f)(\xi). (t-a)^\alpha,
\end{equation}
where $a\leq\xi\leq x$, $\forall x \in (a,b]$.\\

\noindent\textbf{Corollary 1}~~~ Suppose $f(t)\in C[a,b]$ and $^{c}_{t_0} D^{\alpha}_{t}f(t)\in C(a,b)$, $0<\alpha \leq 1$. If $^{c}_{t_0} D^{\alpha}_{t}f(t)\geq 0, \forall t\in (a,b)$, then $f(t)$ is a non-decreasing function for all $t\in[a,b]$; and if $^{c}_{t_0} D^{\alpha}_{t}f(t)\leq 0, \forall t\in (a,b)$, then $f(t)$ is a non-increasing function for all $t\in[a,b]$. \\

\noindent\textbf{Lemma 2} \cite{Kilbas06}~~~The solution to the cauchy problem
$$^{c}_{t_0} D^{\alpha}_{t}x(t)  = \lambda x(t) + f(t),$$
$$x(a)                         =  b (b\in \Re)$$
with $0< \alpha \leq 1$ and $\lambda\in\Re$ has the form
\begin{equation}\nonumber
x(t) = bE_{\alpha}[\lambda(t-a)^{\alpha}] + \int_a^t (t-s)^{\alpha-1} E_{\alpha,\alpha}[\lambda(t-s)^{\alpha}]f(s)ds,
\end{equation}
while the solution to the problem
$$^{c}_{t_0} D^{\alpha}_{t}x(t)  = \lambda x(t),$$
$$x(a)                         =  b ~(b\in \Re)$$
is given by
\begin{equation}\nonumber
x(t) = bE_{\alpha}[\lambda(t-a)^{\alpha}].
\end{equation}

\noindent\textbf{Lemma 3} \cite{Hong-Li16}~~~Let $u(t)$ be a continuous function on $[t_{0},\infty)$ and satisfying
$$^{c}_{t_0} D^{\alpha}_{t}u(t) \leq -\lambda u(t) + \mu,$$
$$u(t_{0})                         =  u_{t_{0}} $$
where $0< \alpha<1$, $(\lambda,\mu)\in\Re^{2}$, $\lambda\neq0$ and $t_{0}\geq0$ is the initial time. Then its solution has the form
\begin{equation}\nonumber
u(t) = \bigg(u_{t_{0}}- \frac{\mu}{\lambda}\bigg)E_{\alpha}[-\lambda(t-t_{0})^{\alpha}] + \frac{\mu}{\lambda}.
\end{equation}

\noindent\textbf{Lemma 4} \cite{LiChen10}~~~Consider the system
\begin{equation}\nonumber
^{c}_{t_{0}} D^{\alpha}_{t} x(t) = f(t,x), t>t_{0}
\end{equation}
with initial condition $x_{t_{0}}$, where $0<\alpha\leq1$, $f:[t_{0},\infty)\times\Omega\rightarrow\Re^{n}$, $\Omega\in\Re^{n}$. If $f(t,x)$ satisfies the locally Lipschitz condition with respect to $x$, then there exists a unique solution of the above system on $[t_{0},\infty)\times\Omega$.
\section{Well-posedness}
\subsection{Non-negativity and boundedness}
Considering biological significance of the problem, we are only interested in solutions that are non-negative and bounded. Denote $\Re^3{}_{+} = \{x\in\Re^{3}|x\geq0 \}$ and $x(t) = (S(t),I(t),P(t))^{T}$.\\

\noindent\textbf{Theorem 3.}~~All solutions of the system (\ref{Eco-epidemiological fractional order model}) which start in $\Re^3{}_{+}$ are non negative and uniformly bounded.\\

\noindent\textbf{Proof.} First we prove that $S(t)\geq0, \forall t\geq0$ assuming $S(0)>0$ for $t=0$. Let us suppose that $S(t)\geq0, \forall t\geq0$ is not true. Then there exists some $t_{1}> 0$ such that
$S(t)> 0$ for $0\leq t< t_{1}$, $S(t) = 0$ at $t = t_{1}$ and $S(t)< 0$ for $t> t_{1}$.\\
From the first equation of (\ref{Eco-epidemiological fractional order model}), we have
\begin{equation}
^{c}_{0} D^{\alpha}_{t_{1}} S(t)|_{t=t_{1}} = 0.
\end{equation}
According to Lemma $1$, we the have $S(t_{1}^{+}) = 0$, which contradicts the fact $S(t_{1}^{+}) < 0$, i.e. $S(t)< 0$ for $t> t_{1}$. Therefore, we have $S(t)\geq0, \forall t\geq0$. Using similar arguments, we can prove $I(t)\geq0, \forall t\geq0$ and $P(t)\geq0, \forall t\geq0$. Next we show that all solutions of system (\ref{Eco-epidemiological fractional order model}) which start in $\Re^3{}_{+}$ are uniformly bounded.\\ We define a function
\begin{equation}
V(t) = S + I + \frac{m}{\theta}P,
\end{equation}
Taking its fractional time derivative, we have
\begin{equation}\nonumber
^{c}_{0} D^{\alpha}_{t} V(t) = rS\bigg(1-\frac{S+I}{K}\bigg) - \mu I - \frac{md}{\theta} P.
\end{equation}
Now for each $\eta>0$, we have
\begin{equation}\label{Boundedness}
\begin{split}
^{c}_{0} D^{\alpha}_{t} V(t) + \eta V(t) = & rS\bigg(1-\frac{S+I}{K}\bigg) - \mu I - \frac{md}{\theta} P + \eta S +\eta I + \frac{m\eta}{\theta} P \\
\leq & -\frac{r}{K}S^{2} + (r + \eta)S + (\eta - \mu) I+ (\eta - d)\frac{md}{\theta}P \\
\leq & \frac{K}{4r}(r + \eta)^{2} + (\eta - \mu) I+ (\eta - d)\frac{md}{\theta}P.
\end{split}
\end{equation}
Taking $\eta < min(\mu, d)$, we have
\begin{equation}
\begin{split}
^{c}_{0} D^{\alpha}_{t} V(t) + \eta V(t) \leq l,
\end{split}
\end{equation}
where $l=\frac{K}{4r}(r + \eta)^{2}>0.$ Applying Lemma $3$, one gets
\begin{equation}
\begin{split}
V(t) \leq & (V(0) - \frac{l}{\eta})E_{\alpha}[-\eta t^{\alpha}] + \frac{l}{\eta}, \\
\leq & V(0) E_{\alpha}[-\eta t^{\alpha}] + \frac{l}{\eta}(1 - E_{\alpha}[-\eta t^{\alpha}]).
\end{split}
\end{equation}
Thus,$V(t)\rightarrow \frac{l}{\eta}$ as $t\rightarrow \infty$ and $0<V(t)\leq \frac{l}{\eta}$. Hence all solutions of system (\ref{Eco-epidemiological fractional order model}) that starts from $\Re^3{}_{+}$ are confined in the region $\Omega = \{(S,I,P)\in\Re^3{}_{+}| ~V(t)\leq \frac{l}{\eta} + \epsilon$, for any $\epsilon>0\}$.
\subsection{Existence and uniqueness}
Now, we  study the existence and uniqueness of the solution of system (\ref{Eco-epidemiological fractional order model}) in the region $\Omega \times[0, T]$, where $\Omega = \{(S,I,P)\in\Re^{3}|~ max\{|S|, |I|, |P|\} \leq M\}$, $T<\infty$ and $M$ is sufficiently large. Denote $X = (S,I,P)$, $\bar{X} = (\bar{S}, \bar{I}, \bar{P})$. Consider a mapping $H(X) = (H_{1}(X), H_{2}(X), H_{3}(X))$ and
\begin{eqnarray}\label{Existence}
H_{1}(X) & = & rS\bigg(1-\frac{S+I}{K}\bigg) - \lambda IS, \nonumber \\
H_{2}(X) & = & \lambda IS-\frac{mIP}{a+I} - \mu I, \\
H_{3}(X) & = & \frac{\theta IP}{a+I}-dP. \nonumber
\end{eqnarray}
For any $X, \bar{X} \in \Omega$, it follows from (\ref{Existence}) that
\begin{equation}\nonumber
\begin{split}
\parallel H(X) - H(\bar{X})\parallel =& \mid H_{1}(X)-H_{1}(\bar{X})\mid +\mid H_{2}(X)-H_{2}(\bar{X})\mid +\mid H_{3}(X)-H_{3}(\bar{X})\mid \\
=& \mid rS\bigg(1-\frac{S+I}{K}\bigg) - \lambda IS - r\bar{S}\bigg(1-\frac{\bar{S}+\bar{I}}{K}\bigg) + \lambda \bar{I}\bar{S}\mid \\
& +\mid \lambda IS-\frac{mIP}{a+I} - \mu I - \lambda \bar{I}\bar{S}+\frac{m\bar{I}\bar{P}}{a+\bar{I}} + \mu \bar{I}\mid \\
& +\mid \frac{\theta IP}{a+I}-dP-\frac{\theta \bar{I}\bar{P}}{a+\bar{I}}+d\bar{P}\mid \\
=& \mid r(S-\bar{S}) - \lambda(IS -\bar{I}\bar{S}) - \frac{r}{K}(S^{2} -\bar{S}^{2} +IS - \bar{I}\bar{S})\mid \\
& +\mid \lambda(IS - \bar{I}\bar{S}) - \mu(I - \bar{I}) - m\bigg(\frac{IP}{a+I} - \frac{\bar{I}\bar{P}}{a+\bar{I}}\bigg)\mid \\
& + \mid \theta\bigg(\frac{IP}{a+I} - \frac{\bar{I}\bar{P}}{a+\bar{I}}\bigg) - d(P - \bar{P})\mid \\
\leq& r\mid S-\bar{S}\mid + \lambda\mid IS -\bar{I}\bar{S}\mid + \frac{r}{K}\mid S^{2} -\bar{S}^{2}\mid +\mid IS - \bar{I}\bar{S}\mid \\
& + \lambda\mid IS - \bar{I}\bar{S}\mid + \mu \mid I - \bar{I}\mid + m\mid\frac{IP}{a+I} - \frac{\bar{I}\bar{P}}{a+\bar{I}}\mid \\
& + \theta \mid\frac{IP}{a+I} - \frac{\bar{I}\bar{P}}{a+\bar{I}}\mid + d\mid P - \bar{P}\mid \\
\leq& \bigg(r +\frac{2rM}{K} + (2\lambda + \frac{r}{K})M\bigg)\mid S - \bar{S}\mid \\
& + \bigg((2\lambda + \frac{r}{K})M + \mu + \frac{aM(m+\theta)}{(a+M)^2}\bigg)\mid I-\bar{I}\mid \\
& + \bigg(\frac{aM(m+\theta)}{(a+M)^2}+d +\frac{M^{2}(m+\theta)}{(a+M)^2} \bigg)\mid P-\bar{P}\mid \\
\leq& L\parallel (S,I,P) - (\bar{S}, \bar{I}, \bar{P})\parallel \\
\leq& L\parallel X - \bar{X} \parallel,
\end{split}
\end{equation}
where $L = max\{r +\frac{2rM}{K} + (2\lambda + \frac{r}{K})M,(2\lambda + \frac{r}{K})M + \mu + \frac{aM(m+\theta)}{(a+M)^2}, \frac{aM(m+\theta)}{(a+M)^2}+d +\frac{M^{2}(m+\theta)}{(a+M)^2}\}$.\\ Thus $H(X)$ satisfies Lipschitz condition with respect to $X$ and it follows from Lemma $4$ that there exists a unique solution $X(t)$ of the system (\ref{Eco-epidemiological fractional order model}) with the initial condition $X(0) = (S(0), I(0), P(0))$.
\section{Dynamical behavior}
To obtain equilibrium points of (\ref{Eco-epidemiological fractional order model}), we solve the following simultaneous equations:
\begin{eqnarray}\label{Equilibrium system}
^{c}_{0} D^{\alpha}_{t}S & = & 0\nonumber \\
^{c}_{0} D^{\alpha}_{t}I & = & 0, \\
^{c}_{0} D^{\alpha}_{t}P & = & 0. \nonumber
\end{eqnarray}
We thus obtain $E_{0} = (0,0,0)$ as the trivial equilibrium, $E_{1} = (K,0,0)$ as the axial equilibrium, $E_{2} = (S_1, I_1, 0)$ as the planar equilibrium, where $S_1 = \frac{\mu}{\lambda}$ and $I_1 = \frac{r(\lambda K-\mu)}{\lambda(r + \lambda K)}$ and $E^{*} = (S^{*},I^{*},P^{*})$ as the interior equilibrium, where
\begin{eqnarray}\label{Equilibrium relation}
S^{*} =  K-\bigg(1 + \frac{\lambda K}{r}\bigg)I^{*},~~ I^{*} =  \frac{ad}{\theta -d},~~ P^{*} = \frac{(a+I^{*})(\lambda S^{*} - \mu)}{m}.
\end{eqnarray}
Note that the equilibria $E_{0}$ and $E_{1}$ always exist. The planar equilibrium point $E_{2}$ exists if $R_{0}>1$, where $R_{0} =\frac{\lambda K}{\mu}$. The interior equilibrium $E^{*}$ exists if $(i)$ $R_{0}>1$ and $(ii)$ $\theta > \theta_{1}$, where $\theta_1=d + \frac{\lambda ad(r + \lambda K)}{r(\lambda K - \mu)}$.\\
The jacobian matrix of system (\ref{Eco-epidemiological fractional order model}) evaluated at $E_{0}$ is given by
\[
J(E_{0}) = \begin{pmatrix}
r & 0 & 0 \\ 0 & -\mu & 0 \\ 0 & 0 & -d \end{pmatrix}.\]
The eigenvalues can be determined by solving the characteristic equation $det(J(E_{0} - \xi I_{3})) = 0$ and they are $\xi_{1} = r ~(>0)$, $\xi_{2} = -\mu ~(<0)$ and $\xi_{3} = -d ~(<0)$. Note that $\mid arg(\xi_{1})\mid = 0$, $\mid arg(\xi_{2})\mid = \pi $, and $\mid arg(\xi_{3})\mid = \pi $. Since the first eigenvalue $\xi_{1}$ does not satisfy $\mid arg(\xi_{1})\mid >\frac{\alpha \pi}{2}$ for all $\alpha\in (0,1]$, therefore $E_{0} = (0, 0, 0)$ is always unstable.\\

\noindent The jacobian matrix $J(E_{1})$ is computed as
\[
J(E_{1}) = \begin{pmatrix}
-r & -r-\lambda K & 0 \\ 0 & \lambda K - \mu & 0 \\ 0 & 0 & -d \end{pmatrix}.\]
The corresponding eigenvalues are $\xi_{1} = -r ~(<0)$, $\xi_{2} = \lambda K - \mu$, $\xi_{3} = -d<0$. Here two cases arise depending on whether $R_0>1$ or $R_0<1$. \\
\noindent\textbf{Case 1:} If $R_0<1$ then we can see that $\mid arg(\xi_{i})\mid  = \pi > \frac{\alpha \pi}{2}, \forall \alpha\in (0,1]$, $i = 1, 2, 3$. Therefore, the equilibrium $E_{1}$ is locally asymptotically stable.\\
\noindent\textbf{Case 2:} If $R_0>1$ then it is easy to see that $\mid arg(\xi_{2})\mid = 0$. In this case, $E_{1}$ is unstable. \\

Performing similar calculations, one can show that the characteristic equation of the Jacobian matrix $J(E_{2})$ can be expressed as
\begin{equation}
\bigg(\xi- (\frac{\theta r (\lambda K - \mu)}{a\lambda(\lambda K + r)+ r(\lambda K-\mu)} - d)\bigg)( \xi^{2} + A\xi + B)=0,
\end{equation}
where $A = \frac{r\mu}{\lambda K} > 0$ and $B = \frac{r\mu (\lambda K- \mu)}{\lambda K} > 0$.
Therefore, one eigenvalue is $\xi_{1} = d_{1} - d$, where $d_{1} = \frac{\theta r (\lambda K - \mu)}{a\lambda(\lambda K + r)+ r(\lambda K-\mu)}$ and the other two are given by $\xi_{2, 3} = \frac{1}{2}(- A \pm \sqrt{A^{2} - 4B})$. Following two cases may arise. \\

\noindent\textbf{Case $d > d_1$:} If $d > d_{1}$ then the equilibrium $E_{2}$ is locally asymptotically stable. Again if $1<R_0<1+\frac{r}{4}$ then $A^2 - 4B>0$. In this case, $\xi_{1}$, $\xi_{2}$, $\xi_{3}$ all are real negative and $\mid arg(\xi_{i})\mid  = \pi > \frac{\alpha \pi}{2}, \forall \alpha\in (0,1]$, $i = 1, 2, 3$. Therefore, the equilibrium $E_{2}$ is stable node if $d > d_{1}$ and $1<R_0<1+\frac{r}{4}$. However, $R_0>1+\frac{r}{4}$ gives $A^2 - 4B<0$. Then $\xi_{1}<0$ and $\xi_{2}, \xi_{3}$ become complex conjugate with negative real parts. Thus, $\mid arg(\xi_{1})\mid  = \pi > \frac{\alpha \pi}{2}$ and $\mid arg(\xi_{2,3})\mid  = \arctan(\frac{\sqrt{-(A^{2} - 4B)}}{A})> \frac{\alpha \pi}{2}$, $\forall \alpha\in (0,1]$. Therefore, the equilibrium $E_{2}$ is stable focus if $d > d_{1}$ and $R_0>1+\frac{r}{4}$.\\
\noindent\textbf{Case $d < d_{1}$:} If $d < d_{1}$ then $E_{2}$ is always unstable. It will be unstable node if $1<R_0<1+\frac{r}{4}$ and unstable focus if $R_0>1+\frac{r}{4}$.\\

\noindent For the interior equilibrium $E^{*}$, the Jacobian matrix is evaluated as
\[\mathbf{J(E^{*})} = \begin{pmatrix}
-\frac{r S^{*}}{ K} & -(\frac{r}{K}+\lambda)S^{*} & 0 \\ \lambda I^{*} & \frac{mI^{*}P^{*}}{(a + I^{*})^{2}} & -\frac{md}{\theta} \\ 0 & \frac{a \theta P^{*}}{(a + I^{*})^{2}} & 0
\end{pmatrix}.\]
The eigenvalues are the roots of the cubic equation
\begin{equation}\label{Cubic equation}
F(\xi) = \xi^{3} + A_{1} \xi^{2} + A_{2} \xi + A_{3} = 0,
\end{equation}
where $A_{1} = \frac{r S^{*}}{K} - \frac{mI^{*}P^{*}}{(a + I^{*})^{2}},~
A_{2} = \frac{amdP^{*}}{(a+I^{*})^{2}} + \frac{r \lambda I^{*}S^{*}}{K} + \lambda^{2} I^{*}S^{*} - \frac{rmS^{*}I^{*}P^{*}}{K(a+I^{*})^{2}},~
A_{3} = \frac{rmdaS^{*}P^{*}}{K(a+I^{*})^{2}}.$

\noindent The discriminant $D(F)$ of the cubic polynomial $F(\xi)$ is
\[
\mathbf{D(F)} = - \begin{vmatrix}
1 & A_{1} & A_{2} & A_{3} & 0 \\ 0 & 1 & A_{1} & A_{2} & A_{3} \\ 3 & 2A_{1} & A_{2} & 0 & 0 \\ 0 & 3 & 2A_{1} & A_{2} & 0 \\ 0 & 0 & 3 & 2A_{1} & A_{2}
\end{vmatrix}.
\]
On expansion, one gets $D(F) = 18A_{1}A_{2}A_{3} + (A_{1}A_{2})^2 - 4A_{3}A_{1}^{3} - 4A_{2}^{3} - 27A_{3}^{2}$. We have the following proposition. \\
\noindent\textbf{Proposition 1.}\begin{itemize}
	\item[(i)] If $D(F) > 0$, $ A_{1}>0$, $A_{3}>0$ and $A_{1}A_{2}- A_{3}>0$ then the interior equilibrium $E^{*}$ is locally asymptotically stable.
	\item[(ii)] If $D(F) < 0$, $A_{1}\geq 0$, $A_{2}\geq 0$, $A_{3} > 0$ and $0< \alpha < \frac{2}{3}$ then the interior equilibrium $E^{*}$ is locally asymptotically stable.
	\item[(iii)] If $D(F) < 0$, $A_{1} < 0$, $A_{2} < 0$ and $\alpha > \frac{2}{3}$ then the interior equilibrium $E^{*}$ is unstable.
	\item[(iv)] If $D(F) < 0$, $A_{1} > 0$, $A_{2} > 0$, $A_{1}A_{2} = A_{3}$ and $0< \alpha< 1$ then the interior equilibrium $E^{*}$ is locally asymptotically stable.
\end{itemize}
{\bf Proof.} ~(i) If $D(F)$ is positive then all the roots of (\ref{Cubic equation}) are real and distinct. If not, let us assume that $F(\xi) = 0$ has one real root $\xi_1$ and another two complex conjugate roots $\xi_2$, $\xi_3$. In terms of the roots, the discriminant of $F(\xi)$ can be written as \cite{Janson07}
\begin{equation}\label{Discriminant eqn}
D(F) =[( \xi_1 - \xi_2)(\xi_1 - \xi_3)(\xi_2 - \xi_3)]^2.
\end{equation}
Note that
\begin{eqnarray}
\begin{split}
( \xi_1 - \xi_2)(\xi_1 - \xi_3)(\xi_2 - \xi_3) = & ( \xi_1 - \xi_2)(\xi_1 - \overline{\xi_2})(\xi_2 - \overline{\xi_2})\\
= & ( \xi_1 - \xi_2)(\xi_1 - \overline{\xi_2})2 Im(\xi_2)i\\
= & ( \xi_1 - \xi_2)\overline{(\xi_1 - \xi_2)}2 Im(\xi_2)i\\
= & 2|\xi_1 - \xi_2|^2 Im(\xi_2)i.
\end{split}
\end{eqnarray}
Thus, \begin{equation}
D(F) =[2|\xi_1 - \xi_2|^2 Im(\xi_2)i]^2 < 0,
\end{equation}
which contradicts the fact that $D(F) > 0$. Therefore, whenever $D(F)>0$ then $F(\xi) = 0$ has three real distinct roots. Since $ A_{1}>0$, $A_{3}>0$ and $A_{1}A_{2}- A_{3}>0$, all roots of $F(\xi) = 0$ has negative real roots or complex conjugate roots with negative real parts. As $D(F) > 0$, so all roots of $F(\xi) = 0$ are real negative. Consequently, $\mid arg(\xi_{i})\mid  = \pi > \frac{\alpha \pi}{2}, \forall \alpha\in (0,1]$, $i = 1, 2, 3$, and the equilibrium $E^{*}$ is  locally asymptotically stable. This completes the proof of (i).\\

\noindent (ii) We have seen in (i) that $F(\xi) = 0$ has one real and two complex conjugate roots if $D(F) < 0$. Since $A_{3} > 0$, following (\ref{Cubic equation}), the real root is negative. We thus consider the roots as $\xi_{1} = -b$, $(b\in R_+)$ and $\xi_{2,3} = \beta \pm i \gamma,$ $(\beta, \gamma \in R)$ and
$$F(\xi) = (\xi +b)(\xi - \beta - i \gamma)(\xi - \beta + i \gamma).$$ Comparing this with (\ref{Cubic equation}), we have
$A_{1} =  b-2\beta,
A_{2} = \beta^2 + \gamma^2 - 2b\beta,
A_{3} = b(\beta^2 + \gamma^2).$
Now $A_{1} \geq 0 \Rightarrow b \geq 2\beta$. Noting $\beta^2 \sec^2\theta= \beta^2 + \gamma^2$ and $A_{2} \geq 0$, we have $sec^2\theta \geq  4.$ Therefore, $\theta= |arg(\xi)| \geq \frac{\pi}{3}$. Since $0< \alpha < \frac{2}{3}$, then $ |arg(\xi)| = \theta \geq \frac{\pi}{3} > \frac{\alpha \pi}{2}$ holds. Thus, all roots of (\ref{Cubic equation}) satisfy $\mid arg(\xi_{i})\mid >\frac{\alpha \pi}{2}, \forall \alpha\in (0,1]$ and the equilibrium $E^{*}$ is locally asymptotically stable. This completes the proof of (ii). Proof of $(iii)$ is similar to the proof of (ii) and hence omitted.\\

\noindent Since $D(F) < 0$, $A_{1} > 0$, $A_{2} > 0$, from the previous case, we have the
\begin{eqnarray}\nonumber
A_{1} =  b-2\beta, ~A_{2} =  \beta^2 + \gamma^2 - 2b\beta, ~A_{3} =  b(\beta^2 + \gamma^2).
\end{eqnarray}
Note that $A_{1} > 0 \Rightarrow b> 2\beta$, $A_{2} > 0 \Rightarrow \beta^2 + \gamma^2 - 2b\beta > 0$
and $A_{1}A_{2} = A_{3} \Rightarrow (b-2\beta)(\beta^2 + \gamma^2 - 2b\beta) = b(\beta^2 + \gamma^2) \Rightarrow \beta(b^2 + \beta^2 + \gamma^2 - 2b\beta) = 0$. Then two cases arise: \\
\noindent\textbf{Case 1:} If $\beta = 0$ then three roots $\xi_{1},\xi_{2},\xi_{3}$ of (\ref{Cubic equation}) are $-b, \pm i\gamma$. One can see that $\mid arg(\xi_{1})\mid  = \pi > \frac{\alpha \pi}{2}$ and $\mid arg(\xi_{2,3})\mid  = \frac{\pi}{2} > \frac{\alpha \pi}{2}, \forall\alpha\in (0,1)$. Therefore, the equilibrium $E^{*}$ is locally asymptotically stable.\\
\noindent\textbf{Case 2:} If $b^2 + \beta^2 + \gamma^2 -2b\beta=0$ then we have $b=\beta$ and $\gamma=0$. Using it in $b> 2\beta$ and $\beta^2 + \gamma^2 >2b\beta$, we obtain $b<0$, which contradicts the assumption $b\in R_+$.\\
Thus, if $A_{1} > 0$, $A_{2} > 0$, $A_{1}A_{2} = A_{3}$ then one root is real negative and the other two are purely imaginary and therefore $\mid arg(\xi_{1})\mid  = \pi > \frac{\alpha \pi}{2}$ and $\mid arg(\xi_{2,3})\mid  = \pi/2 > \frac{\alpha \pi}{2},$ $\forall \alpha\in (0,1)$, implying local asymptotic stability of $E^{*}$. This completes the proof.
\section{Global asymptotic stability}
We now prove the global stability of different equilibrium points of the system (\ref{Eco-epidemiological fractional order model}). \\

\noindent\textbf{Lemma 5} \cite{Vargas15}~~~Let $x(t)\in \Re_{+}$ be a continuous and derivable function. Then for any time instant $t>t_{0}$
\begin{equation}\nonumber
^{c}_{t_{0}} D^{\alpha}_{t}\bigg[x(t) - x^{*} - x^{*}ln\frac{x(t)}{x^{*}}\bigg] \leq \bigg(1-\frac{x^{*}}{x(t)}\bigg)~~{^{c}_{t_{0}}} D^{\alpha}_{t}x(t), x^{*}\in \Re_{+}, \forall \alpha\in(0,1).
\end{equation}

\noindent\textbf{Theorem 4:} The axial equilibrium $E_{1}=(K,0,0)$ is global asymptotically stable if $R_{0} < 1$. \\

\noindent {\bf Proof.} We consider the following Lyapunov function
\begin{equation}
V(S, I, P) = \bigg(S - K - Kln\frac{S}{K}\bigg) + I+ \frac{m}{\theta}P.
\end{equation}
Here $V$ is a $C^1$ function such that $V > 0$ for all values of $(S(t), I(t), P(t))\neq (K,0,0)$ and $V = 0$ only at $(K,0,0)$. Calculating the $\alpha$ order derivative of $V(S, I, P)$~ along the solutions of (\ref{Eco-epidemiological fractional order model}) and using the Lemma $4$, we have
\begin{eqnarray}\label{Globality_1}
\begin{aligned}
^c_{0}D^{\alpha}_{t}V(S, I, P) \leq& \frac{(S - K)}{S} ~{^c_{0}}D^{\alpha}_{t}S(t) + {^c_{0}}D^{\alpha}_{t}I(t)+\frac{m}{\theta} ~{^c_{0}}D^{\alpha}_{t}P(t)\\
=& (S - K)[r(1-\frac{S+I}{K}) - \lambda I] + [\lambda IS-\frac{mIP}{a+I} - \mu I]+ \frac{m}{\theta}[\frac{\theta IP}{a+I}-dP] \\
=& (S - K)[r(1-\frac{S+I}{K})] - \lambda I(S - K)+ [\lambda IS-\frac{mIP}{a+I} - \mu I]+ [\frac{m IP}{a+I}-\frac{dm}{\theta}P] \\
\leq & (S - K)[r(1-\frac{S}{K})] - \lambda I(S - K)+ [\lambda IS-\frac{mIP}{a+I} - \mu I]+ [\frac{m IP}{a+I}-\frac{dm}{\theta}P] \\
=& -\frac{r}{K}(S - K)^{2} + (\lambda K - \mu)I - \frac{dm}{\theta}P. \\
\end{aligned}
\end{eqnarray}
Note that if $R_{0} < 1$ then $^c_{0}D^{\alpha}_{t}V(S,I,P) \leq0, \forall (S, I, P)\in R^{3}_{+}$, and $^c_{0}D^{\alpha}_{t}V(S,I,P) = 0$ at $E_1$. Therefore, the only invariant set on which $^c_{0}D^{\alpha}_{t}V(S,I,P) = 0$ is the singleton $\{E_{1}\}$. Then using Lemma $4.6$ in \cite{Huo15}, which generalizes the integer-order LaSalle's Invariance Principle to fractional-order system, it follows that every nonnegative solution tends to $E_{1}$ when $R_{0} < 1$. Thus, $E_{1}$ is global asymptotically stable if $R_{0} < 1$. \\

\noindent\textbf{Theorem 5:} The planner equilibrium $E_{2} = (S_1, I_1, 0)$ is global asymptotically stable if $d > d_{2}$, where $d_{2} = \frac{\theta r (\lambda K - \mu)}{a\lambda(r+ \lambda K)}$.\\

\noindent {\bf Proof.} Let us define the Lyapunov function as
\begin{equation}
V(S, I, P) = \bigg(S - S_{1} - S_{1}ln\frac{S}{S_{1}}\bigg) + \bigg(I - I_{1} - I_{1}ln\frac{I}{I_{1}}\bigg)+ \frac{m}{\theta}P.
\end{equation}
Here $V$ is a $C^1$ function such that $V > 0$ for all values of $(S(t), I(t), P(t)) \neq (S_1, I_1, 0)$~ and $V = 0$ only at $(S(t), I(t), P(t))= (S_1, I_1, 0)$. As before, we have
\begin{eqnarray}
\begin{aligned}
^c_{0}D^{\alpha}_{t}V(S, I, P) \leq& \frac{(S - S_{1})}{S}~ {^c_{0}}D^{\alpha}_{t}S(t) + \frac{(I - I_{1})}{I} ~{^c_{0}}D^{\alpha}_{t}I(t) + \frac{m}{\theta} ~{^c_{0}}D^{\alpha}_{t}P(t)\\
=& (S - S_{1})[r(1-\frac{S+I}{K}) - \lambda I] + (I - I_{1})[\lambda S-\frac{mP}{a+I} - \mu]+ \frac{m}{\theta}[\frac{\theta IP}{a+I}-dP] \\
=& (S - S_{1})[r(\frac{S_{1}+I_{1}}{K}-\frac{S+I}{K}) - \lambda (I - I_{1})] + (I - I_{1})[\lambda (S -S_{1})-\frac{mP}{a+I}] + [\frac{mIP}{a+I}-\frac{dm}{\theta}P] \\
=& -\frac{r}{K}(S - S_{1})^{2} - \frac{r}{K}(S - S_{1})(I - I_{1}) + [\frac{m I_{1}}{a +I} - \frac{dm}{\theta}]P \\
\leq & -\frac{r}{K}(S - S_{1})^{2} - \frac{r}{K}[\frac{(S - S_{1})^{2} + (I - I_{1})^{2}}{2}] + [\frac{m I_{1}}{a +I} - \frac{dm}{\theta}]P \\
= & -\frac{3r}{2K}(S - S_{1})^{2} - \frac{r}{2K}(I - I_{1})^{2} + [\frac{m I_{1}}{a +I} - \frac{dm}{\theta}]P \\
\leq & -\frac{3r}{2K}(S - S_{1})^{2} - \frac{r}{2K}(I - I_{1})^{2} + [\frac{m I_{1}}{a} - \frac{dm}{\theta}]P.\\
\end{aligned}
\end{eqnarray}
One can easily show that if $d >d_2$, where $d_2= \frac{\theta r (\lambda K - \mu)}{a\lambda(r+ \lambda K)}$ then $^c_{0}D^{\alpha}_{t}V(S,I,P) \leq0, \forall (S, I, P)\in R^{3}_{+}$, and $^c_{0}D^{\alpha}_{t}V(S,I,P) = 0$ at $E_2$. Therefore, the only invariant set on which $^c_{0}D^{\alpha}_{t}V(S,I,P) = 0$ is the singleton $\{E_{2}\}$. Following Lemma $4.6$ in \cite{Huo15}, it follows that if $E_{2}$ exists and $d > d_{2}$ then it is global asymptotically stable.  \\

\noindent \textbf{Remark:}
It is to be noted that $d_{2} > d_{1}$, where $d_{1} = \frac{\theta r (\lambda K - \mu)}{a\lambda(\lambda K + r)+ r(\lambda K-\mu)}$. This shows that global stability of $E_{2}$ implies its local stability but the converse is not necessarily true. There may exist some parametric space where $E_{2}$ is only locally stable.\\

\noindent\textbf{Theorem 6:} The positive interior equilibrium $E^{*}=(S^*, I^*, P^*)$ is global asymptotically stable if ~$\theta_{1} < \theta < \theta_{2}$, where $\theta_{1} = d + \frac{\lambda ad(r + \lambda K)}{r(\lambda K - \mu)}$ and $\theta_{2} = \frac{mdK}{2K(\lambda S^{*} - \mu) - r}$. \\
{\bf Proof.} To prove global stability of $E^*$, we define the Lyapunov function as
\begin{equation}
V(S, I, P) = \bigg(S - S^{*} - S^{*}ln\frac{S}{S^{*}}\bigg) + \bigg(I - I^{*} - I^{*}ln\frac{I}{I^{*}}\bigg)+ \frac{m}{\theta}\bigg(P - P^{*} - P^{*}ln\frac{P}{P^{*}}\bigg),
\end{equation}
where $V$ is a $C^1$ function such that $V > 0$ for all values of $(S(t), I(t), P(t)) \neq (S^{*},I^{*},P^{*})$ and $V = 0$ only at $(S(t), I(t), P(t)) = (S^{*},I^{*},P^{*})$. We then have
\begin{eqnarray}\nonumber
\begin{aligned}
^c_{0}D^{\alpha}_{t}V(S, I, P) \leq& \frac{(S - S^{*})}{S} ~{^c_{0}}D^{\alpha}_{t}S(t) + \frac{(I - I^{*})}{I} ~{^c_{0}}D^{\alpha}_{t}I(t)+\frac{m}{\theta}\frac{(P - P^{*})}{P} ~{^c_{0}}D^{\alpha}_{t}P(t)\\
=& (S - S^{*})[r(1-\frac{S+I}{K}) - \lambda I] + (I - I^{*})[\lambda S-\frac{mP}{a+I} - \mu]+ \frac{m}{\theta}(P - P^{*})[\frac{\theta I}{a+I}-d] \\
=& (S - S^{*})[r(\frac{S^{*}+I^{*}}{K}-\frac{S+I}{K}) - \lambda (I - I^{*})] + (I - I^{*})[\lambda (S -S^{*})-\frac{mP}{a+I}+ \frac{mP^{*}}{a+I^{*}}]\\
&+(P - P^{*})[\frac{mI}{a+I}-\frac{mI^{*}}{a+I^{*}}]\\
=& -\frac{r}{K}(S - S^{*})^{2} - \frac{r}{K}(S - S^{*})(I - I^{*}) - \frac{m(I - I^{*})(PI^{*} - P^{*}I)}{(a + I)(a + I^{*})}\\
=& -\frac{r}{K}(S - S^{*})^{2} - \frac{r}{K}(S - S^{*})(I - I^{*}) + \frac{mP^{*}(I - I^{*})^{2}}{(a + I)(a + I^{*})}
- \frac{mI^{*}(I - I^{*})(P - P^{*})}{(a + I)(a + I^{*})}\\
\leq & -\frac{r}{K}(S - S^{*})^{2} - \frac{r}{K}[\frac{(S - S^{*})^{2} + (I - I^{*})^{2}}{2}] + \frac{mP^{*}(I - I^{*})^{2}}{(a + I)(a + I^{*})}\\
&- \frac{mI^{*}}{(a + I)(a + I^{*})}[\frac{(I - I^{*})^{2} + (P - P^{*})^{2}}{2}]\\
= & -\frac{3r}{2K}(S - S^{*})^{2} + [\frac{mP^{*}}{(a + I)(a + I^{*})} - \frac{mI^{*}}{2(a + I)(a + I^{*})} - \frac{r}{2K}](I - I^{*})^{2} - \frac{mI^{*}(P - P^{*})^{2}}{2(a + I)(a + I^{*})}\\
\leq & -\frac{3r}{2K}(S - S^{*})^{2} + [\frac{mP^{*}}{(a + I^{*})} - \frac{mI^{*}}{2(a + I^{*})} - \frac{r}{2K}](I - I^{*})^{2} - \frac{mI^{*}}{2(a + I^{*})}(P - P^{*})^{2}.\\
\end{aligned}
\end{eqnarray}
Observe that $[\frac{mP^{*}}{(a + I^{*})} - \frac{mI^{*}}{2(a + I^{*})} - \frac{r}{2K}] < 0$ if $\theta < \theta_{2}$, where $\theta_{2} = \frac{mdK}{2K(\lambda S^{*} - \mu) - r}$. In this case, $^c_{0}D^{\alpha}_{t}V(S,I,P) \leq0, \forall (S, I, P) \in R^{3}_{+}$ and $^c_{0}D^{\alpha}_{t}V(S,I,P) = 0$ at $(S^*, I^*, P^*)$. Therefore, the only invariant set on which $^c_{0}D^{\alpha}_{t}V(S,I,P) = 0$  is $\{E^{*}\}$. Following Lemma $4.6$ in \cite{Huo15}, whenever the interior equilibrium $E^{*}$ exists and $\theta_{1} < \theta < \theta_{2}$, where $\theta_{1} = d + \frac{\lambda ad(r + \lambda K)}{r(\lambda K - \mu)}$, $\theta_{2} = \frac{mdK}{2K(\lambda S^{*} - \mu) - r}$ then it is global asymptotically stable.
\section{Numerical Simulations}
In this section, we perform extensive numerical computations of our system (\ref{Eco-epidemiological fractional order model}) for different fractional orders $0 < \alpha \leq 1$. We employ Adamas-type predictor corrector method for our fractional order differential equation (FODE) \cite{Diethelm02, Diethelm04}. We first
replace the FODE system (\ref{Eco-epidemiological fractional order model}) by the equivalent fractional order integral system
\begin{eqnarray}\label{Eco-epidemiological fractional integral eqn}
S(t) & = & S(0) + D^{-\alpha}_{t} \bigg(rS[1-\frac{S+I}{K}] - \lambda IS \bigg), \nonumber \\
I(t) & = & I(0) + D^{-\alpha}_{t} \bigg(\lambda IS-\frac{mIP}{a+I} - \mu I \bigg), \\
P(t) & = & P(0) + D^{-\alpha}_{t} \bigg(\frac{\theta IP}{a+I}-dP \bigg), \nonumber
\end{eqnarray}
and then apply the PECE (Predict, Evaluate, Correct, Evaluate) method. With the following three examples we substantiate our analytical findings.

\noindent \textbf{Example 1:}
We consider the parameter values as $r = 2.0$, $K = 40.0$, $\lambda = 0.015$, $m = 0.52$, $\mu = 0.28$, $a = 15.0$, $\theta = 0.189$, $d = 0.09$ and initial point $S(0) = 30, I(0) = 5, P(0) = 10$. Most of the parameter values are taken from \cite{ChattoBairagi01}. Step size for all simulations is considered as $0.05$. We compute that $D(F) = 0.0077 > 0$, $ A_{1} = 1.0879 >0$, $A_{3} =  0.0028>0$, $A_{1}A_{2}- A_{3} = 0.2909>0$. Thus, following Proposition 2(i), the interior equilibrium $E^{*}$ is stable for $0 < \alpha \leq 1$. In Fig. 1 we plot the solutions of FODE system (\ref{Eco-epidemiological fractional order model}) with different values of $\alpha$. It shows that all populations remain stable for all values of $\alpha$ though solutions reach to equilibrium value more slowly for smaller value of $\alpha$.
\begin{figure}[H]
	\includegraphics[width=6.5in, height=2in]{Shubhojit_Fig_1.eps}\\
	\caption{Asymptotically stable solutions of $S$, $I$ and $P$ populations with different fractional orders $ 0< \alpha < 1$ and standard order $\alpha = 1$. Here $r = 2.0$, $K = 40.0$, $\lambda = 0.015$, $m = 0.52$, $\mu = 0.28$, $a = 15.0$, $\theta = 0.189$, $d = 0.09$.}
	\label{Stable_1.eps}
\end{figure}
For the above parameter values, following Theorem 6, we determine the critical values of the parameter $\theta$ as $\theta_{1} = 0.1723$ and $\theta_{2} = 0.8044$. Fig. 2 demonstrates that solutions starting from different initial values converge to the equilibrium point $E^{*} = (35.7195, 3.2927, 8.9983)$ for $\theta = 0.5$, depicting its global stability.
\begin{figure}[H]
	\centering
	\includegraphics[width=5in, height=2in]{Shubhojit_Fig_2.eps}
	\caption{Trajectories with different initial values converge to the coexistence equilibrium $E^{*}$. These figures indicate the global stability of the equilibrium $E^{*}$ for $(a)$ $\alpha = 0.85$ and $(b)$ $\alpha = 0.95$. Here $\theta = 0.5$ and other parameters are as in Fig. 1.}
	\label{Stable_phase_1.eps}
\end{figure}
If we consider $K = 200, \lambda = 0.15, a = 5.0$ and $\theta = 0.9$, keeping other parameter values unchanged, we notice that the conditions of Proposition 2(iii) are satisfied with $D(F) = -463.8995 <0$, $ A_{1} = -0.9276<0$, $A_{2} = -0.5775<0$. Therefore, the interior equilibrium point $E^{*}$ is unstable for  $\alpha > \frac{2}{3}$ (Fig. 3).

\begin{figure}[H]
	\centering
	\includegraphics[width=6.5in, height=2in]{Shubhojit_Fig_3.eps}\\
	\caption{Unstable solutions of $S$, $I$ and $P$ populations for $\alpha = 0.85$. Here $K = 200, \lambda = 0.15, a = 5.0$, $\theta = 0.9$ and other parameters are as in Fig. 1.}
	\label{Unstable.eps}
\end{figure}

\noindent \textbf{Example 2.}
For a lower value $\theta= 0.08$, we compute $R_{0} = \frac{\lambda K}{\mu} = 2.142> 1$ and $d - d_1 = 0.0025 > 0$. Therefore, following Theorem 5, the equilibrium point $E_{2} = (18.67, 16.4, 0)$ is stable. Time series solutions and phase portraits of the system (\ref{Eco-epidemiological fractional order model}) for different orders are presented in Figure 4 to illustrate the system behavior. Time evolutions (upper panel) show that solutions converge to the equilibrium faster for higher order and phase diagrams (lower panel) indicate that all trajectories with different initial conditions converge to the predator-free equilibrium $E_{2}$, depicting its is global asymptotic stability.\\

\noindent \textbf{Example 3.}
We now consider the same parameter values and initial point as in Ex. 1 except $\lambda = 0.005$. In this case $R_{0} = \frac{\lambda K}{\mu} = 0.7143< 1$ and we observe that all trajectories with different initial conditions converge to the equilibrium $E_{1} = (40, 0, 0)$ (Fig. 5), following Theorem 4. This indicates that the predator-free and infection-free equilibrium $E_{1}$ is globally asymptotically stable for different orders.

\begin{figure}[H]
	\centering

	\includegraphics[width=6in, height=2.5in]{Shubhojit_Fig_4a.eps}\\
	\includegraphics[width=6in, height=2.5in]{Shubhojit_Fig_4b.eps}
	\caption{Upper panel shows time behavior and lower panel shows phase behavior of solutions of system (\ref{Eco-epidemiological fractional order model}) for different orders. These figures indicate global stability of predator-free equilibrium $E_{2}$ for different orders. All parameters are as in Fig. 1.}
	\vspace{2in}
	\label{Stable_phase_3.eps}
\end{figure}
\begin{figure}[H]
	\centering
	\includegraphics[width=6in, height=2.5in]{Shubhojit_Fig_5a.eps}\\
	\includegraphics[width=6in, height=2.5in]{Shubhojit_Fig_5b.eps}
	\caption{Upper panel shows time behavior and lower panel shows phase behavior of solutions of system (\ref{Eco-epidemiological fractional order model}) for different orders. These figures indicate global stability of the equilibrium $E_{1}$ for different orders. Here $\lambda = 0.005$ with other parameters as in Fig. 1.}
	\label{Stable_phase_2.eps}
\end{figure}

\section{Discussion}
In recent past, eco-epidemiological models have received tremendous attention of modelers because these models consider the issues of ecology and epidemiology simultaneously. Various continuous-time models \cite{Arino99,Ventu94} and discrete-time models \cite{Hu17, Hu14, Adak15} have been proposed and analyzed considering different attributes of the eco-epidemiological system. In this paper, we consider an ecological system where prey population grows logistically and predator population feeds on it following type II response function. When prey is infected by some micro-parasites, it is assumed that predator consumes infected prey only as they are weaken by the disease and can not escape predation. This eco-epidemiological situation has been modeled by a system of fractional order nonlinear differential equations. We prove that the solution of this model system exists uniquely and all solutions remain positive and bounded whenever they start with positive initial value, thus justifying the well-posedness of a biological model. We showed that our system contains four equilibrium points. The trivial equilibrium point is always unstable, implying that all populations can not go to extinction simultaneously. The infection- free and predator-free equilibrium is locally and globally asymptotically stable if $R_0 < 1$ and the dynamics is independent of the order of the differential equation. The predator-free equilibrium is locally asymptotically stable for all order if $1<R_0<1+\frac{r}{4}$ and $d>d_1$. However, if the death rate of predator is very high ($d>d_2>d_1$) then $E_2$ is globally asymptotically stable whenever it exists. The coexistence or interior equilibrium exists if $R_0>1$ and $\theta>\theta_1$. By using stability analysis of fractional order system, we have given different sufficient conditions on the system parameters to prove local stability and instability of $E^*$ for different values of the order, $\alpha$. If, however, $\theta_1<\theta<\theta_2$ then the interior equilibrium is globally asymptotically stable for any $\alpha(0<\alpha\leq1)$ whenever it exists. Numerical examples are presented in support of our analytical results. It is observed that solutions converge to the respective equilibrium more slowly as the order of the differential equation becomes smaller, though the qualitative nature of the solutions remain unchanged.\\

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Half of Europeans want to see more naturally flavoured energy drinks
October 21st, 2015October 21st, 2015
Consumers' heightened awareness of artificial flavours and colourings has been putting growing pressure on the European energy drinks market. Indeed, new research from Mintel has found that naturally positioned energy drinks have grown tremendously popular over the last few years and are most sought after by the youngest and oldest consumers in Europe.
The research reveals that today, over half (53%) of German consumers say that they would like to see a wider variety of energy drinks made with natural flavours and colourings, while four in 10 (41%) say they are willing to pay more for varieties containing only natural energy boosters. In other European markets the situation is similar, with 54% of Italian and Spanish consumers and 61% of Polish consumers agreeing that they would like to see more energy drinks with natural ingredients.
And it seems that those aged 16-24 and 55 and over are driving the demand for naturally flavoured energy drinks. In Germany, some six in 10 (61%) consumers aged 16-24 and 60% of respondents aged 55+ would welcome energy drinks with more natural flavours and colours. In Italy, this number is even higher; almost seven out of ten (68%) Italian consumers aged 55 and older say they are keen on natural energy drinks.
This development is reflected by new products that have entered the European market in recent years. Last year one in four (24%) energy drinks launched in Europe claimed to be 'natural,' compared to only one in five (17%) in 2012.
The energy drinks market is one of the fastest growing beverage markets in Europe. Last year, Western Europeans* consumed roughly 1.3 billion litres of energy drinks, with the UK (545 million litres) and Germany (262 million litres) accounting for more than half of the consumption. This is set to further increase, with Western Europeans forecast to drink an approximate 1.8 billion litres of energy drinks in 2019 – an impressive 38% growth in consumption.
Julia Buech, Senior Food & Drink Analyst at Mintel, says:
"Even though traditional brands continue to dominate the fast-growing energy drinks market, more naturally positioned alternatives have seen growth in Europe in recent years. The fact that all age groups seem to be interested in more natural ingredients means that more holistic energy drinks will capture older consumers while not alienating key younger demographics."
Mintel research also reveals a trend towards the so-called 'energy hybrids' which draw on authentic taste profiles from more everyday drink categories. Tea-based energy drinks in particular hold a strong appeal for the increasingly health-conscious consumer. As many as four in 10 (42%) Germans say that they would like to see a wider variety of energy drinks with tea as a caffeine ingredient. In Italy and Poland interest is even higher with almost half of the respondents agreeing; 47% in Italy and 49% in Poland.
Germany in particular has seen an outstandingly strong rise in beverages with natural, plant-based energy sources such as caffeine derived from tea or yerba mate. In the first eight months of 2015, one in seven (15%) energy drinks launched contained these ingredients, up from just one in 20 (5%) over the same period in 2014.
What's more, over a third (35%) of Germans typically indulge in energy drinks at home, with usage on-the-go (30%) and at work or a place of study (16%) falling behind. The same is true for other countries in Europe, where 31% of Spanish, 23% of Italian and 25% of Polish consumers get their boost from energy drinks at home.
"Consumers increasingly want to feel more energised from products that they already consume habitually in their everyday life, in order to get the energy they need without having to greatly change their lifestyle. It is worth noting that energy drinks, while often promoted as the ultimate active lifestyle product, are – like tea – in fact primarily consumed at home. Energy drinks drawing on health-laden ingredients such as tea have the potential to enhance the everyday appeal of the sector, potentially increasing usage among existing consumers and attracting a wider target group." Buech concludes.
*refers to volume consumption of energy drinks in Austria, Belgium, Denmark, Finland, France, Germany, Greece, Italy, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, Turkey and the UK.
All consumer data used in this press release is based on 2,000 internet users aged 16+.
Press review copies of the research and interviews with Senior Food and Drinks Analyst Julia Buech are available on request from the press office.
Plant-based push: UK sales of meat-free foods shoot up 40% between 2014-19
From meatless steak bakes to plant-based Whoppers, it seems that vegan food product launches are now in full swing as new research from Mintel reveals a nation hungry ...
Mintel announces four US Foodservice Trends for 2020
Mintel, the experts in what consumers want and why, has today (January 15, 2020) announced four trends set to impact the US restaurant industry in 2020, bringing new ...

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See Hillary Clinton Belt It Out in The Daily Show's 'Song for Women 2017'
See Hillary Clinton "take it home" for Daily Show correspondents Dulcé Sloan and Desi Lydic after they sing about the Women's March, the sexual assault reckoning and other historic accomplishments by women in 2017
By Tierney McAfee
Sen. John McCain recently called on 2016 Democratic presidential nominee Hillary Clinton to "shut up" after her election loss. But now she's doing the exact opposite as she sings at the top of her (or someone's, anyway) lungs in a show-stealing cameo in The Daily Show's new "Song for Women 2017."
See Clinton "take it home" for Daily Show correspondents Dulcé Sloan and Desi Lydic after they sing about the Women's March, the sexual assault reckoning and other historic accomplishments by women in 2017 — including Beyoncé, Elizabeth Warren, Megyn Kelly and Wonder Woman. (Because, "Who knew you could direct when you have boobies?")
Check out the full video above.


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The Weeknd all of a sudden announces he's bringing his Starboy Tour to NZ!
Out of nowhere, The Weeknd has put out a post, titled "Phase 2" announcing that he's bring his Starboy World Tour to New Zealand.
The Canadian singer, songwriter, and record producer will be making his way down under in November, playing one New Zealand show before jumping the ditch to play 5 Australian shows.
The only details we know so far is that he'll be playing in Auckland on the 29th of November, so mark that one down in your calendar.


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Nothrus taisetsuensis är en kvalsterart som beskrevs av K. Fujikawa 1999. Nothrus taisetsuensis ingår i släktet Nothrus och familjen Nothridae. Inga underarter finns listade i Catalogue of Life.

Källor

Spindeldjur
taisetsuensis

redpajama_set_name: RedPajamaWikipedia


Meet Annabelle who hopefully will steal your heart enough to give her a loving home! She is entirely hand ~ crafted and does indeed have a shy side as reflected in her innocent face. Annabelle also comes with this wonderful tin bucket decorated with prairie flowers and gourd eggs.
My inspiration for this gathering came from a tin maple syrup bucket that was found in a Missouri Barn. It was one of the many items in a large box containing similar Karo Syrup containers however this one was missing the label. The bucket is in good condition as there are no holes, creases or dents to report. The aged tin boasts a dark pewter color that just enhances the overall look of this gathering. There are some areas shown in images 4 & 5 that have light surface rust. The lid is also in good condition and was placed inside the opening to act as a back drop for the rabbit. The wire handle was inserted through a tin tab located on each side of the bucket to secure it.
I was able to find a fabulous Easter Egg Dye Label that I reproduced and hand ~ stitched onto aged burlap. This colorful and graphic label graces the front of the tin and ties in the Easter theme perfectly boasting a rabbit holding an Easter Egg with a chicken off to the left. Different colored eggs dot the landscape with baby chicks hatching all over the place! Cute!
Annabelle was fashioned from an old wool quilt that has been aged for a primitive look. She was completely hand ~ stitched and seems quite content set among-st an array of prairie flowers that include Sweet Annie, Broom Corn and dried Crepe Myrtle. A couple of gourd eggs were also added inside the bucket, one large and one small and look primitively perfect!
This Easter gathering will bring springtime charm to your homestead and Annabelle promises to do her best to make you happy for years to come! Enjoy!
The bucket measures 8 1/4" Tall which includes the tin lid, 6" without & 5" Across the bottom. Annabelle stands 5 3/4" Tall & 2 1/2" @ the Widest point which is at the Bottom.


Why are academics so obsessed with posters?
I've been involved in higher education for some time now – both as a student and an employee – and still to this day it surprises me just how obsessed some people are with posters.
And by posters here I don't mean posters to stick on the wall; no, rather I mean posters to send out by email or for users to download from social media or a website. Got a conference coming up? Make a poster and send it to your mailing lists. Got an event? Make a poster and upload it to your blog.
Posters are not SEO-friendly. Their content is not searchable online.
Errors are difficult to correct. If you have a change of date or change of venue, you can't just go out and edit all the posters you've already sent out into the ether.
Posters are not mobile-friendly. Large files are slow to download and impact on a user's data allowance. They also don't make for easy reading on a mobile device.
Posters can be time-consuming to create. They are not an efficient means of communication and can often be overlooked by a time-starved audience.


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Graph-theoretic approach to stability of multi-group models with dispersal
DCDS-B Home
The stability of bifurcating steady states of several classes of chemotaxis systems
January 2015, 20(1): 249-257. doi: 10.3934/dcdsb.2015.20.249
On conformal measures of parabolic meromorphic functions
Zuxing Xuan 1,
Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing Union University, Beijing, 100101, China
Received January 2014 Revised March 2014 Published November 2014
We prove the absolute continuity of the Hausdorff measure with respect to any conformal measure. These results extend Denker and Urbanski's work on parabolic rational functions.
Keywords: conformal measures, singular values, Parabolic meromorphic function, Julia set, Hausdorff dimension..
Mathematics Subject Classification: Primary: 30B20, 30D05, 30D10; Secondary: 34M05, 37F1.
Citation: Zuxing Xuan. On conformal measures of parabolic meromorphic functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 249-257. doi: 10.3934/dcdsb.2015.20.249
J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548. doi: 10.1090/S0002-9947-1993-1107025-2. Google Scholar
M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc., 43 (1991), 107-118. doi: 10.1112/jlms/s2-43.1.107. Google Scholar
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J. Kotus and M. Urbański, Conformal, geometric and invariant measures for transcendental expanding functions, Math. Ann., 324 (2002), 619-656. doi: 10.1007/s00208-002-0356-y. Google Scholar
J. Kotus and M. Urbański, Fractal measures and ergodic theory of transcendental meromorphic functions, in Transcendental Dynamics and Complex Anaysis, London Math. Soc. Lecture Note Ser., {348}, Cambridge Univ. Press, Combridge, 2008, 251-316. doi: 10.1017/CBO9780511735233.013. Google Scholar
B. O. Stratmann and M. Urbański, The geometry of conformal measures for parabolic rational maps, Math. Proc. Cambridge Phil. Soc., 128 (2000), 141-156. doi: 10.1017/S0305004199003837. Google Scholar
M. Urbański and A. Zdunik, The parabolic map $1/e e^z$, Indag. Math. (N.S.), 15 (2004), 419-433. doi: 10.1016/S0019-3577(04)80009-0. Google Scholar
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Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993
François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
Yan Huang. On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2395-2409. doi: 10.3934/dcds.2018099
Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647
Sabyasachi Mukherjee. Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2565-2588. doi: 10.3934/dcds.2017110
Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
Juan Wang, Yongluo Cao, Yun Zhao. Dimension estimates in non-conformal setting. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3847-3873. doi: 10.3934/dcds.2014.34.3847
Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
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Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure & Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164
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Zuxing Xuan


2019 Events We are planning on participating in ** or sponsoring *** the following for 2019... Events with an * are tentative at this time!!
YES - Volvos and Saab's on their lawn We are planning on holding a cookout for all the participants!!, to be held on the museum grounds as the event comes to an end, should be a great day for all to enjoy!!
Sat. 9/14 3 PM - ?
We haven't decided where at this point in time but Stay Tuned!!!
This years nod is to the Subaru STI for 25 years of production, VW Jetta for 40 years of Production and Citroen's 100 Years of Production. The VCOA is usually the largest - or one of the largest car clubs there. Great show, camping available at the show. VCOA or VCNA usually does a BBQ there. If you need motel rooms, you need to get one early as most are sold out months in advance.
The San Luis Obispo Chapter is hosting this years meet. If your thinking of going, the club intentions are to have you make reservations at the place of your choice, as this is not a "weekend event". The SLO chapter has included several on the VCOA site, as they are not promoting one central location - of which there are many and wineries galore!!


In addition to the 1/35 scale Vietnam U.S. Marines figures in fighting poses; this time BRAVO-6 added a different subject to the series. The kit B6-35032 U.S. Military Cameraman , Nam '68 depicts an unarmed cameraman with his Arriflex-16 camera.
I have to admit that ; besides his incredible sculpting skill, I really appreciate Vladimir Demchenko's style on making detailed researchs on books, movies and internet before sculpting his figures. He catches very interesting points and carries them to his figures. This figure is inspired from a screen shot from another legendary Vietnam movie – Full Metal Jacketdirected by Stanley Kubrick.
The figure is sculpted by Vladimir Demchenko and comes in 5 medium grey resin parts. Sculpting and casting quality are very good. The figure parts can be easily removed from the casting blocks, the junctions are well placed not to damage the parts when removing. There will no need for a serious cleaning process. There were no casting mistakes, air bubbles or excess resin on the sample.
The main part is Body with head and legs. The figure is posed slightly leaning forward as taking a film. Facial details and hair are well defined. Left eye of the figure is closed as the right eye is focused on the viewer. He wears typical jungle fatigue with 4 flip-top cover pockets jacket and trousers with 2 flip-top cover cargo pockets on the sides. As the footgear, he has leather-canvas jungle boots. Over the uniform, he has a M1952 Flakvest. This jacket is developed during Korean War and saw continuous service during the Vietnam War despite being officially superseded by the improved M69 armor. It is collarless and has two cargo pockets in front, a zip front closure and adjustable elastic draw cords on each side. The jacket also had grenade hangers over both cargo pockets, the left one of which contained a pen compartment. Also worn by the AFV crew and Airmen, it was capable to stop shrapnel, rocket fragments, snake bites, fire ant bites but not a bullet. Details of the flak vest are very well defined.
He carries a power supply-battery pack attached to his utility belt on the left side. I think some cables should be added between the pack and camera.
Right arm and Left arm are posed to carry the camera. Cloth folds and buttons on the sleeves are well defined.
Camera : He carries a Arriflex 16S-Standard camera which was introduced in 1952. This is a sturdy, rock-steady camera that sounds something like a blender. It has 100-foot capacity, but it is much more rugged and uses a reflex viewfinder. It can use a 400-foot external magazine as seen on this kit and uses a variety of electric motors. Both hands are sculpted on the camera.
Lens of camera : 3 -lens turret of the camera is given as a seperate part.
Interesting subject and great additional figure to Vietnam War scenes. Sculpted in great details and cast in high quality.
Highs: Very nice subject, great sculpt and high quality cast.
Verdict: Great addition to Vietnam War scenes.


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Northern Explorer Northern Explorer
Northland Road Trip - Recommended 10-Day Itinerary Northland Road Trip - Recommended 10-Day Itinerary
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Journey through the winterless north on the Twin Coast Discovery Highway, covering both the east and west coasts and venturing to the very tip of New Zealand. Stunning beaches from the serene to the rugged, rich Maori heritage and majestic forests and marine reserves will keep you captivated every step of the way.
Auckland to Wellsford
Before leaving Auckland, take time to discover Waiheke Island, enjoying its vineyards, secluded beaches, thriving art community, great walks, restaurants and captivating views back to the city. Or jump on a relaxing Auckland Habour Sailing Experience with Explore taking in the volcanic peaks and city skyline from the best view in town.
Departing Auckland head northwest, with the first stop on your journey being Kumeu, home to many of Auckland's great vineyards and restaurants. A little further north is Woodhill Forest, a mountain biker's paradise and equally as great for those on foot. Driving from Helensville to Wellsford on hilly rural roads, you're afforded amazing views over Kaipara Harbour and surrounding farmland.
Wellsford to Opononi
You'll reconnect with State Highway 1 briefly from Wellsford to Brynderwyn before heading off on State Highway 12, again along the west coast. Just south of Brynderwyn is Kaiwaka, a small village known for its Dutch cheese; that's surely worth a stop!
Once you hit the coastline on State Highway 12, you'll be rewarded with view upon view of beautiful sandy beaches. Baylys Beach, just west of Dargaville, is one such example and a great place to spend the day. The crystal clear Kai Iwi Lakes are also a great spot for camping, fishing and hiking.
The Kauri Coast - Waipoua, Mataraua and Waima Forests - are the largest areas of native forest in Northland. Waipoua Forest has some of the largest and oldest living rainforest trees in the world, including Tane Mahuta - 'Lord of the Forest' - New Zealand's largest living Kauri tree, estimated to be 2,000 years old, 51 metres high, with a girth of 13.77 metres! Join Footprints Waipoua for an intimate encounter to learn how these trees are inter-twined with the lives of local Maori and the important role they play in the eco-system that is the Waipoua Forest.
Recommended: Discover the Bay cruise at the Hole in the Rock, Bay of Islands.
Recommended: Coastal Kayakers, Bay of Islands
There are some absolute must-do's when travelling through the Far North, one of these is 90 Mile Beach (Ahipara to Scott Point), which despite being a sandy beach is actually a highway! Sadly you can't drive your rental vehicle on it unfortunately. Interestingly, the beach is only 55 miles long (88km), early missionaries thought it to be a lot longer when they traversed the beach on horseback in 1800s! Grab a boogie board and ride the Te Paki sand dunes. At low tide, dig for Tuatua (a local shellfish delicacy) which sit just below the surface of the sand. Ninety Mile Beach is also a great spot for swimming, fishing and spectacular sunsets.
Back on State Highway 1, drive from Kaitaia to Cape Reinga or Te Rerenga Wairua in Maori, meaning "the leaping off place of spirits" (about 1.5 hours drive). You've now reached the very top of New Zealand, marked by a quaint lighthouse, where the Tasman Sea meets the Pacific Ocean in a swirling collision of oceanic proportions. Five kilometres to the east is Tapotupotu Bay, a great spot for a rest, picnic or camping.
Cape Reinga to Bay of Islands
The east coast of Northland is renowned for its spectacular, unspoilt beaches - Matai Bay, Karikari Peninsula, Doubtless Bay, Taupo Bay, Whangaroa, Matauri Bay and many others. Don't be surprised if you're one of only a few on the beach because there are just so many amazing spaces to choose from! Check out Coca Cola Lake at the end of Ramp Road on Tokerau Beach where the water is the colour of Coca Cola due to the peat that surrounds it. If you're into diving, the Rainbow Warrior (the Greenpeace boat which was sunk in 1985 to form a natural reef), now lies at the Cavalli Islands and is home to amazing marine life. Hungry? Be sure to visit Mangonui for the best fish 'n' chips in New Zealand!
The Bay of Islands encompasses 144 islands as well as Kerikeri, Waitangi, Paihia and Russell and is regarded as a 'game fisher's' and boater's paradise. To truly appreciate this enchanting part of the world take it all in from the end of a 1200ft tow line with Flying Kiwi Parasail, or hit the water and explore the islands with the local experts at Coastal Kayakers. A night or two at the luxurious Waterfront Suites, Bay of Islands, part of Heritage Hotels Boutique Selection, is an absolute must and will make exploring the Bay almost effortless.
Waitangi is the birthplace of New Zealand, home to the historic Treaty House where the Treaty of Waitangi was signed in 1840, marking the beginning of New Zealand as a nation. The historic buildings, treaty house, waka, grounds and surrounding bush walkways are impressive.
Across from Paihia is Russell, the first capital of New Zealand following the signing of the Treaty of Waitangi, before it was relocated to Auckland. Russell is a quaint township, with a cluster of tourist shops, bars and restaurants and home to the oldest licensed pub in New Zealand - The Duke of Marlborough - dating back to 1827. It's an excellent place to watch the sun go down, or enjoy a picnic on the beach directly below. Whilst in Russell, check out Christ Church, New Zealand's oldest church originally built in 1835 and still serving the community today. You can catch a passenger ferry from Paihia to Russell, or a car ferry further south from Opua.
Recommended: Flying Kiwi Parasail, Bay of Islands
Recommended: Footprints Waipoua, Northland
Kawakawa to Whangarei
Be sure to stop in Kawakawa, famous for toilets, pies and steam engines! Austrian artist Frederick Hundertwasser designed and built the famous Kawakawa toilets in the late 1980's; they're a work of art that attract thousands each year. The creamed paua pies at a local bakery are to die for and you just have to meet Gabriel, the last steam engine of her kind still operating in New Zealand. Less than 30 kilometres from Kawakawa heading west, not far from Kaikohe, are the Ngawha Hot Springs, where you can bathe in hot mud thermal pools. A little further on at Gumdiggers Park near Awanui, see 40,000 -150,000 year old massive buried kauri trees that have been exposed. (You can also reach the hot springs and the Gumdiggers Park from the State Highway 12 route).
Driving south from Kawakawa at Waiomio, take a side road to the Kawiti Glow Worm cave with amazing rock formations and lush rainforest.
Once you arrive in Whangarei, drive east to Whangarei Heads and spend a day at Ocean Beach, a favourite spot for surfies. Tutukaka, a little further on, is a marine wonderland and rated by National Geographic as one of the "top coastal destinations on this planet"! White sandy beaches, snorkelling or diving at Poor Knight's Islands Marine Reserve and the world's largest sea cave, Riko Riko.
Whangarei to Auckland
At this point consider an eastern detour to the spectacular beaches of Langs Beach, Waipu Cove, Mangawhai Heads and others. Matakana is a bustling village servicing neighbouring farmland and the popular holiday spot, Omaha. Saturday mornings there's a flurry of activity at the village market, packed with local produce and artisan delights.
Sail where the wind takes us on a 1.5hr sailing experience on Auckland's stunning harbour, taking in the volcanic peaks and city skyline from the best view in town. Enjoy commentary from Explore's knowledgeable crew who will share local insights into the harbour and its history, and you'll even have the opportunity to help the crew sail this 50ft monohull sailing yacht. Discover more.
Since 1988 Coastal Kayakers has been offering guided tours of the historic Waitangi River, Haruru Falls and the crystal clear waters of and alluring beaches of the Bay of Islands. Half Day, Full Day or Multi Day options are available, catering for all levels of experience. Freedom hire is also available for the more adventurous kayaker looking to explore at their own pace. Discover more.
After the exhilaration of launching from the back of our custom-designed Parasailing boat, sit back and relax and soak up the breathtaking aerial views of the Bay of Islands, enjoying the peace and quiet from the end of our 1200ft towline. Flying Kiwi Parasail have been operating safely in the Bay for 18 years, and offer single, tandem or triple flight options - no experience required. Discover more.
Let local guides take you on a journey through nature's stages of evolution, while providing a mythological and interactive interpretation of life in the forest. The spiritual environment of the Waipoua Forest provides a natural stage for unforgettable encounters with some of the largest remaining kauri trees in the world. Discover more.
Recommended Airline
Explore more of the country with Air New Zealand. We make more than 2000 flights a week to 22 unique locations. For a nation of adventurers like ours, that opens up the entire country for countless new experiences. All you need to do is choose your next adventure. Discover more.
Recommended Accommodation
These all-suite luxury Paihia apartments are located only steps away from Paihia's restaurants and wharf,
ideally situated to take advantage of all that the glorious Bay of Islands and Northland region has to offer. Discover more.
Other Recommended Road Trips
Heartland Loop
Northwest Wilds
Eastern Escape
Main Divide
Heritage Trail
Southern Discovery
Kauri Coast


I wrote an earlier post about Charles Lummis and "The Bibliosmiles", an organization designed by group of librarians who probably had too much time on their hands at the 1906 American Library Association Narragansett Pier (RI) Conference. In doing some research on ALA's centennial I came across "ALA Centennial Vignette No. 9" in American Libraries for February, 1976. The vignette was about the Bibliosmiles and was written by Ed Gleaves, at the time Director of the School of Library Science at George Peabody College in Nashville, TN. Gleaves wrote about the first meeting of the Bibliosmiles at the 1907 ALA Conference in Ashville, NC where the organization was officially founded. The organization's motto was "To keep the Book Dust off our Top Shelves" and its password was, "Cheer Up, ALA". Their official seal is shown to the left. The officers, who included some heavy hitters in ALA, gave themselves some interesting titles: John Cotton Dana was the Grand Ha Ha; Tessa L. Kelso was Minehaha; W. P. Cutter was Sardonic Grin; Adelaide R. Hasse was Big Stick; Samuel S. Green was Supt. of Edification; Joseph F. Daniels was Glad Hand; F. K. W. Drury was Subdued Snicker; E. L. Pearson was Main Guy; and Charles F. Lummis was Grim Reality. Grim Reality spoke first at their meeting at the conference according to Gleaves - "Mr. Chairman: In our profession, even more than in most, we must accept Civilization. In our profession, as much as in any, we need to take it with reservation .. We have to catch, ourselves young, and organize against the Habit of Huddling... The whole tendency of civilization is to run together in an indistinguishable mess. We must Get Together, to Keep ourselves Apart... ." Later in his presentation Grim Reality makes a surprisingly serious point for an organization founded in humor, "I wish to call your attention further to the fact that, while ladies have their full share of library positions, they have never been admitted to those serious and subterranean councils in which the real fate of the ALA is determined. With the highest regard for the generic wisdom of this convention, I respectfully submit that there ARE women fully worthy to share our efforts to retain humanity in the libraries... ." In an essay titled "Adelaide Hasse: The New Woman as Librarian" in the book Reclaiming the American Library Past: Writing the Women In edited by Suzanne Hildenbrand (Ablex Publishing, 1996), Clare Beck wrote the following about the participation of Hasse and Kelso in the Bibliosmiles: "Among librarians, Hasse and Kelso's liveliness stood out to an extent that they were chosen as two of the three women members of the Bibliosmiles, a mildly rowdy organization founded by their old friend from Los Angeles, the flamboyant Charles Lummis, as a refuge from the solemnity of ALA meetings. The Bibliosmiles, 'Librarians who are Nevertheless Human' dedicated to 'keeping the dust off our top shelves,' specialized in comic songs and speeches, but the group was also a meeting place for some of the iconoclasts of the library world, men like Lummis and John Cotton Dana, who thought highly of Hasse and reinforced her belief that libraries could be a vital and innovative part of progressive change." Gleaves notes in his article that by 1911, "The Biblioshmiles, being human, were soon to pass into history, and with them went their dream of laughter, their irresistible protest against undue solemnity in the profession of librarianship. A rally of librarians to be remembered, they are largely forgotten - the final grim reality." And so, lest we forget.
Hello! Say, where did you get that Bibliosmiles seal? Was it reproduced in the American Libraries vignette? Thanks!


One of the most exciting and challenging aspects of network and system administration is the constant change in our environment. Continual exposure to new technologies requires us to quickly learn and adapt. Windows 2000 is the newest challenge. The 3rd Large Installation System Administration of Windows NT/2000 conference will give you the expert knowledge and latest tips to migrate to Windows 2000, and also to successfully administer both Windows NT and 2000.
This year's remarkable program offers a variety of tutorials, talks, and informal gatherings for system and network administrators on topics such as Windows 2000 migration, enterprise-level system management, configuration management, interoperability with existing environments, security, administration, and various applications and languages such as Samba, Perl, and SQL.
Keynote Speaker Marcus Ranum, CEO of Network Flight Recorder, Inc., will speak on the challenge of managing security in very complex environments on "Internet time." William (Bill) Murray, CISSP, Information Systems Security Consultant to Deloitte & Touche, will deliver the closing address on managing security in an environment where, although security is absolutely critical, users and managers mostly take note of uptime and availability.
Regardless of the number of clients on your network or the complexity you face, if you currently manage Windows NT or 2000 or will do so in the future, this is the conference you need to attend. Meeting and discussing projects and issues with your peers will certainly spark your own creativity and ingenuity.


Technically, sushi is sushi because of the cooked rice with vinegar. It can be served with any filling (cooked fish, raw fish, or just veggies) and still be sushi. Raw fish seems to be most common, but there are usually options for folks who don't care for that idea.
Technically, sushi is sushi just because it's made with vinegared rice. It can have any filling (raw fish, cooked fish, veggies, fake krab, etc) and still be sushi. Raw fish is most common, but there are usually options for those who don't care for that idea, especially in American restaurants.
Apple, tree, you see the connection.
Haha. When I first read this comment, I was trying to see the connection between a few of Pax's former romantic partners…Apple. Tree.


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Kevin Ledo
Taking Perspective
Type: Oil & Acrylic on Wood Panel
Auth: This RUN comes signed by Kevin Ledo and includes an Original Artwork Letter of Provenance from 1xRUN.
Conjuring Now
This piece of original artwork comes ready to hang.
Kevin Ledo's artistic practice crosses back and forth through the boundaries of mural/street art, fine art, and gold leaf art installations. Ledo aims to create visually bold, and iconic compositions, focusing on portraiture and the human form interlaced with stylized abstract forms. Ledo's figures are mostly void of foreground and background, and are suspended in a minimalistic and slightly surreal world, captured in moments of contemplation, intimacy, emotion and wonderment.
Find more by Kevin Ledo at kevinledo.com
Follow Kevin on Instagram @kevinledo, Instagram @LedoKevin & Facebook
ABOUT Kevin Ledo:
The subject and approach to Ledo's artwork changes according whether it's public art or his personal explorations in fine art. While there is a common visual thread between these different artistic activities, his public art has several different considerations that are taken into account before creating. Some of these considerations include the neighbourhood the mural is set to be in and the history of the area and people who live there. Ledo's public murals are often celebrations of people and diversity and aim to relate to the communities they are found in. Whenever possible, Ledo's own photography is used as reference for his murals. In some cases the people used in Ledo's murals are people from the local community.
Ledo works with a variety of mediums on his murals, including both spray paint and latex wall paint, and on occasion gold leaf. He can work solely with either spray paint or latex wall paint, but usually blend the two techniques into one painting.
MORE FROM Kevin Ledo
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Cross-Discipline Community Discussions
In the past year, we have produced exhibitions, lecture events, and have worked with local groups of eco-conscious individuals, K-12 schools, and Universities, to create activities and interventions aimed at developing simple solutions for a more sustainable life.
The following events were produced by Final Straw, with the help of local organizations and institutions who align with our cause.
Food, Art, and Ethics — Edinburgh, Scotland
We hosted a diverse panel discussion featuring Dr. Emily Brady (Edinburgh University, Professor of Environmental Philosophy), Mike Small (Director of Fife Diet, Europe's largest local food project), and Ben Twist (Director of, Creative Carbon Scotland). These three leading doers and thinkers in three specific areas of environment and sustainability responded to the notion of a holistic view of nature and humanity, one which asks for a closer relationship between people and environment.
This free event was produced with the cooperation of University of Edinburgh, Edinburgh College of Art, and our speakers all graciously donated their time to serve on the panel.
Local food activist, Mike Small speaks at the Panel Discussion at opening night of the Final Straw exhibition, November 26, 2013 at Edinburgh College of Art.
Emily Brady, Ben Twist and Chris Fremantle discuss at opening night of the Final Straw exhibition, November 26, 2013 at Edinburgh College of Art.
Human:Nature — Megijima, Japan
We came together as a community, to learn from some of Japan's most innovative farmers, all of whom approach sustainable agriculture in different ways. Our goal was to better understand how the connection between people and nature can help build a future for our environment and for our own local communities.
The event was produced for the Setouchi International Triennale by Final Straw in partnership with Isao Suizu, Kei Toh, and Ikumasa Hayashi, with funding from Aichi University of the Arts.
Mr. Matsuzawa lectures at the [HUMAN:NATURE] symposium in Megijima, Japan.
Johann Barbie and Songyi Lee discuss the [HUMAN:NATURE] application at the symposium in Megijima, Japan.
Farmer Hiromu Kohno discusses the intersection of farming and art at the [HUMAN:NATURE] symposium in Megijima, Japan.
Open Source Farming and the City — Seoul, South Korea
Our first lecture event to raise awareness and to tell the story of what on earth natural farming has to do with office workers and apartment-tower dwellers. Natural farmer, Ryosok Hong was our keynote speaker.
The event was produced by Final Straw with gracious support from the executive team at Project Noah.


The cost of living comfortably in Haliburton County is the second highest in all of Ontario.
Ontario's Living Wage Network, an organization dedicated to helping communities better understand how much it costs to live without fear of not having enough money, says that the "living wage" for Haliburton County is $19.42 an hour.
Rosie Kadwell, a staff member with the Haliburton, Kawartha, Pine Ridge Health Unit says that the living wage for the area was calculated by the health unit, using the framework provided by the Living Wage Network. Kadwell says the health unit based their calculations on a four-person family with both adults working 37 hours a week. Kadwell says that the framework asks you the cost of rent, transportation, internet usage and hydro. But she adds that the framework does not account for savings, personal debt, or being a homeowner rather than renting.
Kadwell told My Haliburton Now.com's newsroom that the living wage calculator reflects what it costs to thrive in a community. She adds that a living wage means that your basic needs like food, shelter and bills are all taken care of without having to worry that you'll have enough in your account.
Kadwell explained that the living wage calculator is a great tool for a number of reasons, adding that this gives communities insight into how much people need to make to not just live somewhere but also thrive there too. The calculator also helps the individual understand how much they need to make to be able to contribute to the community they live in. According to Kadwell, impoverished people cannot give back to the community because they simply can not afford to do so.


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Why ImmunoGen's Shares Soared 13% Higher Today
By Todd Campbell – Sep 17, 2015 at 2:39PM
NASDAQ: IMGN
(0.85%) $0.04
Price as of January 27, 2023, 4:00 p.m. ET
ImmunoGen is on tap to deliver a presentation to investors tomorrow.
What's happening: Ahead of its investor conference tomorrow, shares in ImmunoGen (IMGN 0.85%) are jumping by 13% today.
Why it's happening: It's been quite a year for ImmunoGen's investors as shares have climbed 197% from their lows last December.
ImmunoGen's rally this year has been due in large part to the selling of its Kadcyla royalty revenue stream to TPG Special Situation Partners for $200 million in cash back in March and rising optimism for its clinical-stage pipeline.
Tomorrow, management will presumably offer up insight into how it plans to deploy its cash stockpile in the coming year to advance promising therapies including IMGN853, a therapy for ovarian cancer that put up impressive phase 1 results in June, into later-stage trials.
Additionally, investors could get updates on a myriad of other clinical stage oncology drugs, many of which are being developed with the help of big pharma collaborators, including Novartis.
However, therapies of particular interest to investors tomorrow include IMGN529, which is being pursued as an adjunct therapy to Rituxan in B-cell malignancies, and IMGN779, a new anticancer medicine that the company has previously indicated it will submit an investigational new drug application for this fall. Updates on those drugs could be particularly insightful given that ImmunoGen hasn't licensed them out to a partner yet.
Overall, since there's likely to be plenty of information presented to digest tomorrow, investors may want to tune in.
Todd Campbell has no position in any stocks mentioned. Todd owns E.B. Capital Markets, LLC. E.B. Capital's clients may have positions in the companies mentioned. The Motley Fool recommends ImmunoGen. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.
IMGN
$4.75 (0.85%) $0.04
ImmunoGen (IMGN) Q2 2022 Earnings Call Transcript
Is ImmunoGen Stock a Bargain After Dropping 39%?
Why Shares of ImmunoGen Jumped 20.3% in December
Social Security Has an Immigration Problem -- and It's Getting Worse


Have you ever seen a pregnant woman and thought she was glowing? Well, animals have a similar effect. Pregnant animals are surprisingly cute, but just like humans, they're miserable when with child. The dichotomy of cute and uncomfortable is obvious in photographs. Here are 20 animals that look totally cute, yet miserable, when pregnant.
Cats are already cute without being pregnant. Who doesn't love a fluffy kitty? But have you ever seen a little cat critter with child? Their bellies get big and hang low. It makes them look especially cute. However, just like pregnant humans, pregnant kitties have to suffer the gestational side effects, too. That pregnant cat is probably sore, tired, and ready to lay down and purr. Cats are just one of many animals that are both cute and miserable when pregnant. Keep reading to find out about the others!


Mulching a garden has many benefits, most we already know, but before you start laying it, think about the different types of mulches and find out the right way to apply them. This can save you time, money, effort and keep your plants happier.
Mulches can be organic or inorganic. What is the difference? Organic mulch is made from compostable materials like straw, cane sugar, barks, composted organic matter, etc. These materials break down after time and provide the soil with extra nutrients to aid plant health and growth. Wood chips such as pine bark and cypress mulch are larger in size and are ideal to let the water penetrate through to the underneath soil. They also take longer to break down and generally easier to spread due to their lightweight properties. Inorganic mulches are generally made from rock and pebbles. Popular rocks used are, river stones, white quartz, gravel and scoria. They don't breakdown and are more suitable for front entrance gardens, small gardens around patios and pot plants to add some design highlights.
How thick do I mulch?
It depends on the material used. If you are using a mulch from packaging, good suppliers will state on the back of the packaging the rate of application. Be careful not to mulch too thickly, the water needs to soak through to the underneath soil. Mulch that is applied too thick will act as a water barrier, water will soak into the mulch and not reach the soil and roots.
Lawn clippings are unpredictable. They can contain grass seed heads and weed seeds. Don't risk it.
Very Important: Keep the mulch away from the base of the plant. Thick mulch up against the plant stem can lead to root rot.
Follow the "rate of application ' directions on the mulch packet. Applying the mulch at the optimum rate will save you money, time and protect and water your plants effectively.
Use a slow release fertiliser, such as Robust and water in to the soil before applying the mulch.
One mulch that has all the right properties like fertiliser, aesthetics and water saving properties is Searles Premium Garden Mulch. This mulch, as well as its appealing features, contains a 4 months feeding fertiliser and double strength Penetraide Re-wetting Granules to help save watering. ALL IN ONE, you can't go wrong.
Copyright © 2017 JC & AT Searle Pty Ltd. All rights reserved.


Individuals can view a complete list of approved nonprofits by using the Organization Decal Guide. The guide also provides a sample of the decal and a contact name for each entity.
If acquired during your regular renewal period, there is not an additional application or fee outside of your regular registration fees. If the plate is aquired outside of your renewal period, there is an additional $10 fee.
The entity is responsible for determining who qualifies for the decal and the fee to be charged. The entity is also responsible for the administration and issuance of the decals. Individuals are not required to have their entity decal on their plates before placing them on their automobile.
Emblem specialty License Plates are different than regular plates because they have a 3" x 3" square space designated for the decals.
Upon sale or transfer of a vehicle bearing emblem specialty plates that display an emblem specialty decal, the plates shall remain with the owner and may be transferred to another vehicle, upon approval by the department.
If you have further questions about the program, please contact the division at (605) 773.3541 (605) 773.3541.


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Lawmaker wants to fine parents of school bullies up to $500
HARRISBURG, Pennsylvania — A Pennsylvania lawmaker wants to fine parents and require them to perform community service if their children get caught bullying others at school.
State Rep. Frank Burns' proposed bill gives parents three strikes. The first time a child is caught bullying someone, the school would be required to notify the parents of how the school handled the situation. For the second offense, the parents would have to take a bullying class and attend a bullying resolution conference. The third time, the parents would receive a court citation and have to pay a fine of up to $500 along with the possibility of performing community service.
"Bullying is underreported and often unaddressed in any meaningful way," Burns said in a statement on his website. "When it's not addressed, bullying can escalate quickly from taunts and hurtful online posts to physical assaults and—in worst cases—suicide. Holding students, parents and officials at all levels accountable is the only way to put an end to this scourge." – READ MORE
IHOP in Maine apologizes after server makes black teens pay upfront for meal
WATCH: President Trump Proposes Creating A 'Space Force'
True Pundit Staff March 14, 2018


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Huntly, Strathbogie and Howe of Alford is one of the nineteen wards used to elect members of the Aberdeenshire Council. It elects four Councillors.

Councillors

Election Results

2022 Election
2022 Aberdeenshire Council election

Robbie Withey was arrested and charged in relation to a disturbance in Alford on 19 April 2022 and as a result, he was suspended from the Conservative Party. He was therefore no longer a Conservative candidate in the Huntly, Strathbogie and Howe of Alford Ward, however, it was too late to remove him from the ballot paper. He was subsequently returned as an Independent Councillor for the ward at the election.

2017 Election
2017 Aberdeenshire Council election

2012 Election
2012 Aberdeenshire Council election

2007 Election
2007 Aberdeenshire Council election

References

Wards of Aberdeenshire


Atrium Health has revealed a data breach which exposed information belonging to roughly 2.65 million patients. "One record accessed is one too many," Atrium Health told us in relation to the breach, which was caused by the organization's billing vendor, a third-party known as AccuDoc Solutions.
Between September 22 and September 29, an unauthorized threat actor was able to gain access to databases containing the records, which included names, home addresses, dates of birth, insurance policy information, service dates, medical record numbers, and account balances. In addition, roughly 700,000 Social Security numbers were exposed. Financial information such as credit card numbers is not thought to be at risk.
Read more about the Atrium Health data breach on ZDNet.
If data breaches were a film genre, third-party cyber-risk would be the talk of producers and casting agents; it's where the money is. Third-party breach scenarios dominate the headlines. The scares are all different — compromised health records, weapons designs, or automakers' trade secrets — but the plot is the same: leaked and stolen files via compromised contractors, supply chains, or business partners.
The ephemeral specter of third-party cyber-risk haunts the C-suite. Leaders complain they can spend untold sums and time ratcheting down their company's internal security measures only to see their data and reputation suffer the consequences of errors and carelessness at other companies.
Read about how to confront third-party risks on DarkReading.
Google announced it is shutting down the Google+ social network after the company's engineers found an API bug that might have exposed some private profile data for more than 500,000 Google+ users. The company said the bug was located in the Google+ People API.
By default, Google+ users can grant access to their profile data to third-party apps. Google+ users can also allow a third-party app to access the public profile information of a user's friends. In a blog post, Ben Smith, Google fellow and vice president of engineering, said the bug allowed third-party apps to also gain access to users' data that was marked private.
Read more about the potential data leak and the end of Google+ on ZDNet.
The retail industry's cybersecurity preparedness continues to lag behind almost every other sector despite efforts by the major credit card associations to bolster retail security via the Payment Card Industry Data Security Standard (PCI DSS).
Third-party risk management firm SecurityScorecard recently analyzed a total of 1,444 domains in the retail industry with an IP footprint of at least 100 and compared the average SecurityScorecard grade of the retail industry to other vertical markets. The exercise showed the retail industry had the second-lowest application security performance among major sectors.
Read more about the findings of the new report on DarkReading.
It isn't just seedy websites putting browsers at risk anymore: A new report out today shows how the state of the Web today has been rocked by the increasingly toxic combination of dynamic content and the use of third-party data sources to serve up that active content.
According to data released earlier this year, the most expensive data breaches start with third parties. Whether it is from poor configuration of online resources managed by a service provider, insecure third-party software, or insecure communication channels with partners, working with third parties can expose organizations to a ton of risks if they don't pay close enough attention.
This year has offered up some crucial examples of the consequences of lax partner and vendor management.
Read about 6 recent examples of third-party breaches on DarkReading.
We all know that an insider threat is often the biggest challenge an organization needs to be equipped to deal with. Some of the most infamous breaches in recent history have been the result of "trusted" insiders who have turned to the dark side. The other, rather obvious threat is from the unknown attackers who are probing our environment looking for weaknesses and exploiting them for fame, money, or just because.
So, is that it? There are only people inside our networks and bad actors outside? Well, according to Joe Campbell, Principal Security Advisor at One Identity, there's a middle ground that we need to remember and that is the 3rd party partner.
Read why Joe Campbell believes that 3rd party partners, who are neither trusted insiders nor external threats, may be your biggest challenge yet, on CSO.
It is difficult to grasp the full extent of it all, no less manage it, especially with the rise of cloud-based services for which most organizations lack any visibility into, understanding of, or control over the development, integration or deployment of the underlying technology.
Read how organizations can approach supply chain risk management by recalling the four pillars of cyber SCRM: security, integrity, resilience and quality, and by considering NIST's five-step approach, on Security Magazine.