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the volterra s model for population growth of a species within a closed system is given in @xcite as @xmath2 where @xmath3 is the birth rate coefficient , @xmath4 is the crowding coefficient and @xmath5 is the toxicity coefficient .
the coefficient @xmath6 indicates the essential behaviour of the population evolution before its level falls to zero in the long term .
@xmath7 is the initial population and @xmath8 denotes the population at time @xmath9 .
this model includes the well - known terms of a logistic equation , and in addition it , includes an integral term @xmath10 that characterizes the accumulated toxicity produced since time zero@xcite .
+ we apply scale time and population by introducing the non - dimensional variables @xmath11 to obtain the non - dimensional problem @xmath12 where @xmath13 is the scaled population of identical individuals at time @xmath9 and @xmath14 is a prescribed non - dimensional parameter .
the only equilibrium solution of eq .
( [ eq3 ] ) is the trivial solution @xmath15 and the analytical solution @xcite @xmath16d\tau ) \label{eq4}\ ] ] shows that @xmath17 for all @xmath9 if @xmath18 .    the solution of eq .
( [ eq1 ] ) has been of considerable concern .
although a closed form solution has been achieved in @xcite , it was formally shown that the closed form solution can not lead to any insight into the behaviour of the population evolution @xcite . some approximate and numerical solutions for volterra s population model
have been reported .
the successive approximations method was suggested for the solution of eq .
( [ eq3 ] ) , but was not implemented . in this case , the solution @xmath13 has a smaller amplitude compared to the amplitude of @xmath13 for the case @xmath19 .    in @xcite , several numerical algorithms namely the euler method , the modified euler method , the classical fourth - order runge - kutta method and runge - kutta - fehlberg method for the solution of eq .
( [ eq3 ] ) are obtained .
moreover , a phase - plane analysis is implemented . in @xcite ,
the numerical results are correlated to give insight on the problem and its solution without using perturbation techniques .
however , the performance of of the traditional numerical techniques is well - known in that it using provides grid points only , and in addition , it requires a large amounts of calculations .    in @xcite , the series solution method and the decomposition method
are implemented independently to eq .
( [ eq3 ] ) and to a related non - linear ordinary differential equation .
furthermore , the padapproximations are used in the analysis to capture the essential behaviour of the populations @xmath13 of identical individuals and approximation of @xmath20 and exact value of @xmath20 for different @xmath21 were compared .
small @xcite solved the volterra s population model by the singular perturbation method .
this author scaled out the parameters of eq .
( [ eq1 ] ) as much as possible and considered four different ways to do this .
he considered two cases @xmath14 small and @xmath14 large .
it is shown in @xcite that for the case @xmath19 , where populations are weakly sensitive to toxins , a rapid rise occurs along the logistic curve that will reach a peak and then is followed by a slow exponential decay . and , for large@xmath21 , where populations are strongly sensitive to toxins , the solutions are proportional to @xmath22 .    in @xcite adomian decomposition method and sinc - galerkin method were compared for the solution of the same integral equation in this paper .
this showed that adomian decomposition method is more efficient and easier to use for the solution of volterra s population model .    in @xcite
the approach is based upon composite spectral functions approximations .
the properties of composite spectral functions consisting of few terms of orthogonal functions utilized to reduce the solution of the volterra s model to the solution of a system of algebraic equations .    in @xcite
rational chebyshev and hermite functions collocation approach were compared for the solution of volterra s population model growth model of a species within a closed system .
they reduced the solution of this problem to the solution of a system of algebraic equations .    in @xcite applied two common collocation approaches based on radial basis functions to solve volterra s population model .    in @xcite a numerical method based on hybrid function approximations
was proposed to solve volterra s population model .
these hybrid functions consist of block - pulse and lagrange - interpolating polynomials .
also , in @xcite volterra s population growth model of a species within a closed system is approximated by collocation method based on two orthogonal functions , sinc and rational legendre functions .
momani et al .
@xcite and xu @xcite used a numerical and analytical algorithm for approximate solutions of a fractional population growth model , respectively .
the first algorithm is based on adomian decomposition method ( adm ) with pad approximants and the second algorithm is based on homotopy analysis method ( ham ) .
many problems in science and engineering arise in infinite and semi - infinite domains .
different numerical methods have been proposed for solving problems on various domains such as fem@xcite , fdm@xcite and spectral@xcite methods and meshfree method@xcite .
the use of the rbf is an one of the popular meshfree method for solving the differential equations @xcite . for many years the global radial basis functions such as gaussian
, multi quadric , thin plate spline , inverse multiqudric and etc was used @xcite .
these functions are globally supported and generate a system of equations with ill - condition full matrix.to convert the ill - condition matrix to a well - condition matrix , csrbfs can be used instead of global rbfs .
csrbfs can convert the global scheme into a local one with banded matrices , which makes the rbf method more feasible for solving large - scale problem @xcite .
the most popular family of csrbf are wendland functions .
this function introduced by holger wendland in 1995 @xcite .
wendland starts with the truncated power function @xmath23 which be strictly positive definite and radial on @xmath24 for @xmath25 , and then he walks through dimension by repeatedly applying the operator i. + * definition @xcite *   with @xmath23 we define @xmath26 it turns out that the functions @xmath27 are all supported on [ 0,1 ] . +
* theorem 1 @xcite *   the function @xmath27 are strictly positive definite ( spd ) and radial on @xmath24 and are of the form @xmath28,\nonumber\\ 0 &        r>1 , \end{cases}\ ] ] with a univariate polynomial @xmath29 of degree @xmath30 .
moreover , @xmath31 are unique up to a constant factor , and the polynomial degree is minimal for given space dimension @xmath32 and smoothness @xmath33 @xcite .
wendland gave recursive formulas for the functions @xmath27 for all @xmath34 .
we instead list the explicit formulas of @xcite . + * theorem 2 @xcite *  the function @xmath27 , @xmath35 have form @xmath36,\nonumber\\ & & \phi_{s,2}\doteq(1-r)_+^{l+2}[(l^2 + 4l+3)r^2+(3l+6)r+3],\nonumber\\ & & \phi_{s,3}\doteq(1-r)_+^{l+3}[(l^3 + 9l^2 + 23l+15)r^3+(6l^2 + 36l+45)r^2\nonumber\\ & & + ( 15l+45)r+15],\nonumber\end{aligned}\ ] ] where @xmath37 , and the symbol @xmath38 denotes equality up to a multiplicative positive constant .
+  the case @xmath39 follows directly from the definition .
application of the definition for the case @xmath40 yields @xmath41,\nonumber\end{aligned}\ ] ] where the compact support of @xmath42 reduces the improper integral to a definite integral which can be evaluated using integration by parts .
the other two cases are obtained similarly by repeated application of @xmath43.@xcite we showed the most of wendland functions in table [ table . 1 . ] .
@*3|c| @xmath27 & smoothness & spd + @xmath44 & @xmath45 & @xmath46 + @xmath47 & @xmath48 & @xmath46 + @xmath49 & @xmath50 & @xmath46 + @xmath51 & @xmath52 & @xmath46 + @xmath53 & @xmath54 & @xmath46 + @xmath55 & @xmath56 & @xmath46 +    [ table . 1 . ]
the one - dimensional function @xmath57 to be interpolated or approximated can be represented by an csrbf as @xmath58 where @xmath59,\nonumber\\ & & \xi = [ \xi_1,\xi_2 , \cdots , \xi_n ] ^t,\nonumber\end{aligned}\ ] ] @xmath60 @xmath61 is the input , @xmath62 is the local support domain and @xmath63s are the set of coefficients to be determined . by using the local support domain
, we mapped the domain of problem to csrbf local domain . by choosing @xmath64 interpolate nodes @xmath65 in domain : @xmath66 to summarize the discussion on the coefficients matrix , we define @xmath67 where : @xmath68^t,\nonumber\\ & & a=[\phi ^t(x_1 ) , \phi ^t(x_2 ) , \cdots , \phi ^t(x_n)]^t\nonumber\\ & & = \begin{pmatrix }   \phi_1(x_1)&\phi_2(x_1)&\cdots & \phi_n(x_1)\cr \phi_1(x_2)&\phi_2(x_2)&\cdots & \phi_n(x_2)\cr \vdots & \vdots & \ddots & \vdots \cr \phi_1(x_n)&\phi_2(x_n)&\cdots & \phi_n(x_n)\nonumber \end{pmatrix}.\end{aligned}\ ] ] note that @xmath69 , by solving the system ( [ system ] ) , the unknown coefficients @xmath63 will be achieved .      in the indirect method , the formulation of the problem starts with the decomposition of the highest order derivative under consideration into crbf .
the obtained derivative expression is then integrated to yield expressions for lower order derivatives and finally for the original function itself .
+ we approximate @xmath70 for solving the model by icsrbf : @xmath71 by using integral operator @xmath72 we have @xmath73 now , to obtain @xmath74 we define the residual functions by substituting eqs .
( [ eq4])-([eq6 ] ) in eq .
( [ eq3 ] ) @xmath75 the set of equations for obtaining the coefficients @xmath74 come from equalizing eq .
( [ eq7 ] ) to zero at @xmath64 interpolate nodes @xmath76 from @xmath77 where @xmath78 is a last interpolate node and @xmath79 is a arbitrary parameter . @xmath80
we applied the method presented in this paper to examine the mathematical structure of @xmath13 .
table ( [ tab1 ] ) shows the maximum of @xmath81 for some @xmath21 and @xmath82 by using in comparison with exact solution and adm solution by wazwaz @xcite .
the resulting graph of eq .
( [ eq3 ] ) is shown in fig .
( [ fig ] ) .    ) for @xmath82 and @xmath83 . ]    @*7|c| @xmath21&@xmath20 & @xmath62&@xmath79&@xmath64&@xmath84&@xmath85 + @xmath86&@xmath87 & @xmath88&@xmath89&@xmath90&@xmath91&@xmath92 + @xmath93&@xmath94 & @xmath88&@xmath95&@xmath96&@xmath97&@xmath98 + @xmath99&@xmath100 & @xmath88&@xmath101&@xmath96&@xmath100&@xmath102 + @xmath103&@xmath104 & @xmath105&@xmath106&@xmath96&@xmath107&@xmath108 + @xmath109&@xmath110 & @xmath105&@xmath111&@xmath112&@xmath113&@xmath114 +    [ tab1 ]    .minimum value of @xmath115 which is obtained with @xmath62 and @xmath79 for icsrbf . [ cols="^,^",options="header " , ]     [ tab2 ]    the local support domain @xmath62 and arbitrary parameter @xmath79 must be specified by the user.an important unsolved problem is to find a approach to determine the optimal size of @xmath62@xcite .
the accuracy of these csrbf depends on the choice of @xmath62 and @xmath79 . by the meaning of residual function in case of eq .
( [ eq7 ] ) , we try to minimize @xmath1 by choosing good @xmath62 and @xmath79 parameters .
we define @xmath1 as @xmath116 were @xmath117 @xmath118 is @xmath119th - order legendre polynomial .
table ( [ tab2 ] ) show the minimum of @xmath1 which is obtained with local support domain @xmath62 and arbitrary parameter @xmath79 .
a method has been presented for solving volterra s population model which is an integro - ordinary differential equation , based on the compactly supported radial basis functions approximation . in this work ,
we applied two common icsrbf methods on the volterra s population model without converting it to an ordinary differential equation .
we used @xmath0 function .
this function are proposed to provide an effective but simple way to improve the convergence of the solution by collocation method .
as appeared from the figures , we have shown that , when the constant @xmath14 is small , this type of population is relatively insensitive to toxins , and when @xmath120 is large , population of this type are extremely sensitive to toxins .
additionally , through the comparison with adm , we have showed that the icsrbf approach have good reliability and efficiency .
in this paper , indirect collocation approach based on compactly supported radial basis function ( csrbf ) is applied for solving volterra s population model . 
 the method reduces the solution of this problem to the solution of a system of algebraic equations . 
 volterra s model is a non - linear integro - differential equation where the integral term represents the effect of toxin . to solve the problem 
 , we use the well - known csrbf : @xmath0 . numerical results and residual norm ( @xmath1 ) 
 show good accuracy and rate of convergence .    
 * keywords : * volterra s population model , compact support radial basis functions , collocation method . + * msc(2010 ) : * primary : 34g20 ; secondary : 34k28 .    corresponding author : member of research group of scientific computing 
 fax : + 98 2122431653 . ]