Mathematical Problems in Engineering

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Volume 2008 |Article ID 945420 | https://doi.org/10.1155/2008/945420

Abdoul R. Ghotbi, A. Barari, D. D. Ganji, "Solving Ratio-Dependent Predator-Prey System with Constant Effort Harvesting Using Homotopy Perturbation Method", Mathematical Problems in Engineering, vol. 2008, Article ID 945420, 8 pages, 2008. https://doi.org/10.1155/2008/945420

Solving Ratio-Dependent Predator-Prey System with Constant Effort Harvesting Using Homotopy Perturbation Method

Academic Editor: Cristian Toma
Received01 Feb 2008
Revised29 Feb 2008
Accepted13 Mar 2008
Published01 Sep 2008

Abstract

Due to wide range of interest in use of bioeconomic models to gain insight into the scientific management of renewable resources like fisheries and forestry, homotopy perturbation method is employed to approximate the solution of the ratio-dependent predator-prey system with constant effort prey harvesting. The results are compared with the results obtained by Adomian decomposition method. The results show that, in new model, there are less computations needed in comparison to Adomian decomposition method.

1. Introduction

Partial differential equations which arise in real-world physical problems are often too complicated to be solved exactly, and even if an exact solution is obtainable, the required calculations may be practically too complicated, or it might be difficult to interpret the outcome. Very recently, some promising approximate analytical solutions are proposed such as Exp-function method, Adomian decomposition method (ADM), variational iteration method (VIM), and homotopy perturbation method (HPM).

HPM is the most effective and convenient method for both linear and nonlinear equations. This method does not depend on a small parameter. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter š‘āˆˆ[0,1], which is considered as a ā€œsmall parameter.ā€ HPM has been shown to effectively, easily, and accurately solve a large class of linear and nonlinear problems with components converging to accurate solutions. HPM was first proposed by He [1ā€“7] and was successfully applied to various engineering problems.

The motivation of this paper is to extend the homotopy perturbation method (HPM) [8ā€“17] to solve the ratio-dependent predator-prey system. The results of HPM are compared with those obtained by the ADM [18]. Different from ADM, where specific algorithms are usually used to determine the Adomian polynomials, HPM handles linear and nonlinear problems in simple manner by deforming a difficult problem into a simple one. The HPM is useful to obtain exact and approximate solutions of linear and nonlinear differential equations.

In this paper, we assume that the predator in model is not of commercial importance. The prey is subjected to constant effort harvesting with š‘Ÿ, a parameter that measures the effort being spent by a harvesting agency. The harvesting activity does not affect the predator population directly. It is obvious that the harvesting activity does reduce the predator population indirectly by reducing the availability of the prey to the predator. Adopting a simple logistic growth for prey population with š‘’>0,š‘>0, and š‘>0 standing for the predator death rate, capturing rate, and conversion rate, respectively, we formulate the problem asš‘‘š‘„š‘‘š‘”=š‘„(1āˆ’š‘„)āˆ’š‘š‘„š‘¦š‘¦+š‘„āˆ’š‘Ÿš‘„,š‘‘š‘¦=š‘‘š‘”š‘š‘„š‘¦š‘¦+š‘„āˆ’š‘’š‘¦,(1.1) where š‘„(š‘”) and š‘¦(š‘”) represent the fractions of population densities for prey and predator at time š‘”, respectively. Equations (1.1) are to be solved according to biologically meaningful initial conditions š‘„(0)ā‰„0 and š‘¦(0)ā‰„0 [18].

2. Applications

In this section, we will apply the HPM to nonlinear differential system of ratio-dependant predator-prey,š»ī€·ī€ø=ī€·ī€øī‚ƒšæī€·šœˆī€øī€·š‘¢šœˆ,š‘1āˆ’š‘āˆ’šæ0ī€øī‚„ī‚ƒš“ī€·šœˆī€øī€·š‘Ÿī€øī‚„ī€ŗī€»+š‘āˆ’š‘“=0,š‘āˆˆ0,1,š‘Ÿšœ€Ī©,(2.1)where š“(šœˆ) is a general differential operator which can be divided into a linear part šæ(šœˆ) and a nonlinear part š‘(šœˆ) and š‘“(š‘Ÿ) is a known analytical function. š‘āˆˆ[0,1] is an embedding parameter, while š‘¢0 is an initial approximation of the equation which should be solved, and satisfies the boundary conditions.

According to the HPM (relation (2.1)), we can construct a homotopy of system as follows: ī€·ī€øī‚€šœˆ1āˆ’š‘2Ģ‡šœˆ1+šœˆ1Ģ‡šœˆ1āˆ’Ģ‡š‘„0y0āˆ’Ģ‡š‘„0š‘„0ī‚ī‚€šœˆ+š‘2Ģ‡šœˆ1+šœˆ1Ģ‡šœˆ1āˆ’ī€·ī€øšœˆ1āˆ’š‘āˆ’š‘Ÿ1šœˆ2+šœˆ2šœˆ21āˆ’ī€·ī€øšœˆ1āˆ’š‘Ÿ21+šœˆ31ī‚ī€·ī€øƗī‚€šœˆ=0,1āˆ’š‘2Ģ‡šœˆ2+šœˆ1Ģ‡š‘£2āˆ’Ģ‡y0y0āˆ’x0Ģ‡y0ī‚ī‚€šœˆ+š‘2Ģ‡šœˆ2+šœˆ1Ģ‡šœˆ2+ī€·ī€øšœˆš‘’āˆ’š‘1šœˆ2+ešœˆ22ī‚=0,(2.2) where dot denotes differentiation with respect to š‘”, and the initial approximations are as follows:š‘£1,0(š‘”)=š‘„0š‘£(š‘”)=š‘„(0),2,0(š‘”)=š‘¦0(š‘”)=š‘¦(0).(2.3) Assume that the solution of (2.2) can be written as a power series in š‘ as follows:šœˆ1=šœˆ1,0+š‘šœˆ1,1+š‘2šœˆ1,2+š‘3šœˆ1,3šœˆ+ā‹Æ,2=šœˆ2,0+š‘šœˆ2,1+š‘2šœˆ2,2+š‘3šœˆ2,3+ā‹Æ,(2.4) where šœˆš‘–,š‘—(š‘–,š‘—=1,2,3,ā€¦) are functions yet to be determined. Substituting (2.3) and (2.4) into (2.2), and arranging the coefficients of p powers, we haveī‚€š‘£2,0Ģ‡š‘£1,0+š‘£1,0Ģ‡š‘£1,0ī‚+ī‚€š‘£31,0āˆ’š‘£21,0+š‘£1,0Ģ‡š‘£1,1+š‘£2,0Ģ‡š‘£1,1+š‘Ÿš‘£1,0š‘£2,0+š‘š‘£1,0š‘£2,0āˆ’š‘£1,0š‘£2,0+š‘£2,0š‘£21,0+š‘Ÿš‘£21,0ī‚š‘+ī€·š‘£1,1Ģ‡š‘£1,1+š‘£1,0Ģ‡š‘£1,2+š‘£2,0Ģ‡š‘£1,2+š‘£2,1Ģ‡š‘£1,1+2š‘Ÿš‘£1,0š‘£1,1+š‘š‘£1,0š‘£2,1+2š‘£2,0š‘£1,0š‘£1,1+š‘Ÿš‘£1,1š‘£2,0+š‘Ÿš‘£1,0š‘£2,1+š‘š‘£1,1š‘£2,0āˆ’š‘£1,0š‘£2,1āˆ’š‘£1,1š‘£2,0+š‘£2,1š‘£21,0āˆ’2š‘£1,0š‘£1,1+3š‘£21,0š‘£1,1ī‚š‘2+ī‚€š‘£1,1Ģ‡š‘£1,2+š‘£1,2Ģ‡š‘£1,1+š‘£1,0Ģ‡š‘£1,3+š‘£2,1Ģ‡š‘£1,2+š‘£2,0Ģ‡š‘£1,3+š‘£2,2Ģ‡š‘£1,1+š‘£2,0š‘£21,1āˆ’š‘£1,0š‘£2,2āˆ’š‘£1,2š‘£2,0āˆ’š‘£1,1š‘£2,1+š‘£2,2š‘£21,0+š‘Ÿš‘£21,1+3š‘£1,0š‘£21,1āˆ’š‘£21,1+š‘š‘£1,1š‘£2,1+š‘š‘£1,0š‘£2,2+š‘š‘£1,2š‘£2,0+š‘Ÿš‘£1,0š‘£2,2+š‘Ÿš‘£1,1š‘£2,1+š‘Ÿš‘£1,2š‘£2,0+2š‘£2,0š‘£1,0š‘£1,2+2š‘Ÿš‘£1,0š‘£1,2+2š‘£2,1š‘£1,0š‘£1,1+3š‘£21,0š‘£1,2āˆ’2š‘£1,0š‘£1,2ī‚š‘3ī‚€š‘£+ā‹Æ=0,2,0Ģ‡š‘£2,0+š‘£1,0Ģ‡š‘£2,0ī‚+ī‚€š‘’š‘£1,0š‘£2,0āˆ’š‘š‘£1,0š‘£2,0+š‘£2,0Ģ‡š‘£2,1+š‘£1,0Ģ‡š‘£2,1+š‘’š‘£22,0ī‚š‘+ī‚€š‘£2,1Ģ‡š‘£2,1+š‘’š‘£1,0š‘£2,1āˆ’š‘š‘£1,0š‘£2,1+š‘’š‘£1,1š‘£2,0āˆ’š‘š‘£1,1š‘£2,0+2š‘’š‘£2,0š‘£2,1+š‘£2,0Ģ‡š‘£2,2+š‘£1,1Ģ‡š‘£2,1+š‘£1,0Ģ‡š‘£2,2ī‚š‘2+ī‚€š‘’š‘£22,1+š‘£2,1Ģ‡š‘£2,2+š‘£2,2Ģ‡š‘£2,1+š‘£2,0Ģ‡š‘£2,3+š‘£1,1Ģ‡š‘£2,2+š‘£1,2Ģ‡š‘£2,1+š‘£1,0Ģ‡š‘£2,3+š‘’š‘£1,0š‘£2,2+š‘’š‘£1,1š‘£2,1āˆ’š‘š‘£1,0š‘£2,2āˆ’š‘š‘£1,1š‘£2,1+š‘’š‘£1,2š‘£2,0āˆ’š‘š‘£1,2š‘£2,0+2š‘’š‘£2,0š‘£2,2ī‚š‘3+ā‹Æ=0.(2.5) In order to obtain the unknown of šœˆš‘–,š‘—(š‘„,š‘”),š‘–,š‘—=1,2,3,ā€¦, we must construct and solve the following system which includes 6 equations, considering the initial conditions of šœˆš‘–,š‘—(0)=0,š‘–,š‘—=1,2,3,ā€¦ :š‘£2,0Ģ‡š‘£1,0+š‘£1,0Ģ‡š‘£1,0š‘£=0,31,0āˆ’š‘£21,0+š‘£1,0Ģ‡š‘£1,1+š‘£2,0Ģ‡š‘£1,1+š‘£1,0š‘£2,0+š‘š‘£1,0š‘£2,0āˆ’š‘£1,0š‘£2,0+š‘£2,0š‘£21,0+š‘Ÿš‘£21,0š‘£=0,1,1Ģ‡š‘£1,1+š‘£1,0Ģ‡š‘£1,2+š‘£2,0Ģ‡š‘£1,2+š‘£2,1Ģ‡š‘£1,1+2š‘Ÿš‘£1,0š‘£1,1+š‘š‘£1,0š‘£2,1+2š‘£2,0š‘£1,0š‘£1,1+š‘Ÿš‘£1,1š‘£2,0+š‘Ÿš‘£1,0š‘£2,1+š‘š‘£1,1š‘£2,0āˆ’š‘£1,0š‘£2,1āˆ’š‘£1,1š‘£2,0+š‘£2,1š‘£21,0āˆ’2š‘£1,0š‘£1,1+3š‘£21,0š‘£1,1š‘£=0,2,0Ģ‡š‘£2,0+š‘£1,0Ģ‡š‘£2,0=0,š‘’š‘£1,0š‘£2,0āˆ’š‘š‘£1,0š‘£2,0+š‘£2,0Ģ‡š‘£2,1+š‘£1,0Ģ‡š‘£2,1+š‘’š‘£22,0š‘£=0,2,1Ģ‡š‘£2,1+š‘’š‘£1,0š‘£2,1āˆ’š‘š‘£1,0š‘£2,1+š‘’š‘£1,1š‘£2,0āˆ’š‘š‘£1,1š‘£2,0+2š‘’š‘£2,0š‘£2,1+š‘£2,0Ģ‡š‘£2,2+š‘£1,1Ģ‡š‘£2,1+š‘£1,0Ģ‡š‘£2,2=0.(2.6) From (2.4), if the first three approximations are sufficient, then setting š‘=1 yields the approximate solution of (1.1) toš‘„(š‘”)=limš‘ā†’1š‘£1(š‘”)=š‘˜=3ī“š‘˜=0š‘£1,š‘˜(š‘”),š‘¦(š‘”)=limš‘ā†’1š‘£2(š‘”)=š‘˜=3ī“š‘˜=0š‘£2,š‘˜(š‘”).(2.7) Therefore,v1,0(š‘”)=š‘„0š‘£(š‘”)=š‘„(0),(2.8)1,1x(š‘”)=āˆ’0ī‚€š‘„20āˆ’š‘„0āˆ’š‘¦0+š‘„0š‘¦0+š‘Ÿš‘¦0+š‘š‘¦0+š‘Ÿš‘„0ī‚š‘”š‘„0+š‘¦0,š‘£(2.9)1,21(š‘”)=2ī‚€š‘„0+š‘¦0ī‚3š‘„ī‚€ī‚€0š‘”2ī‚€3š‘¦0š‘„20āˆ’š‘„20š‘š‘¦0+2š‘„30š‘š‘¦0+3š‘„40š‘Ÿ+6š‘„30š‘¦20āˆ’3š‘¦30š‘„0+š‘„30š‘Ÿ2āˆ’9š‘„30š‘¦0+6š‘„40š‘¦0āˆ’9š‘„20š‘¦20+2š‘¦30š‘„20āˆ’2š‘„30š‘Ÿāˆ’2š‘Ÿš‘¦30āˆ’2š‘š‘¦30+š‘2š‘¦30+š‘Ÿ2š‘¦30+š‘„20š‘š‘¦0š‘Ÿ+3š‘„0š‘Ÿš‘¦20š‘+š‘¦20š‘„0š‘’š‘+š‘š‘„20š‘¦0š‘’āˆ’š‘š‘„20š‘¦0š‘āˆ’3š‘„0š‘š‘¦20+3š‘„0š‘¦20+3š‘¦30š‘„0š‘Ÿ+3š‘¦30š‘„0š‘āˆ’6š‘„20š‘Ÿš‘¦0+2š‘„50āˆ’3š‘„40+š‘¦30+2š‘Ÿš‘¦30š‘+9š‘„30š‘Ÿš‘¦0āˆ’6š‘„0š‘Ÿš‘¦20+9š‘„20š‘¦20š‘Ÿ+5š‘„20š‘¦20š‘+š‘„30+3š‘„0š‘Ÿ2š‘¦20+3š‘„20š‘Ÿ2š‘¦0,š‘£ī‚ī‚ī‚(2.10)2,0(š‘”)=š‘¦0š‘£(š‘”)=š‘¦(0),(2.11)2,1y(š‘”)=0ī‚€āˆ’š‘’š‘„0+š‘š‘„0āˆ’š‘’š‘¦0ī‚š‘”š‘¦0+š‘„0,š‘£(2.12)2,21(š‘”)=āˆ’2ī‚€š‘¦0+š‘„0ī‚3š‘¦ī‚€ī‚€0š‘”2ī‚€3š‘¦0š‘’š‘„20š‘+š‘¦20š‘š‘„0š‘’+2š‘’š‘„30š‘āˆ’š‘š‘„20š‘¦0āˆ’š‘š‘„0š‘¦20āˆ’š‘2š‘„30+š‘š‘„30š‘¦0+š‘š‘„20š‘¦0š‘Ÿ+š‘š‘„0š‘¦20š‘+š‘š‘„20š‘¦20+š‘š‘„0š‘¦20š‘Ÿāˆ’š‘’2š‘„30āˆ’3š‘¦0š‘’2š‘„20āˆ’3š‘¦20š‘’2š‘„0āˆ’š‘¦30š‘’2.ī‚ī‚ī‚(2.13) We also obtained š‘£1,3 and š‘£2,3, but because they were too long to maintain, we skip them and only use them in the final numerical results. In this manner, the other components can be easily obtained by substituting (2.8) through (2.13) into (2.7) as follows:ī‚€xš‘„(š‘”)=š‘„(0)āˆ’0ī‚€š‘„20āˆ’š‘„0āˆ’š‘¦0+š‘„0š‘¦0+š‘Ÿš‘¦0+š‘š‘¦0+š‘Ÿš‘„0ī‚š‘”š‘„0+š‘¦0ī‚+12ī‚€š‘„0+š‘¦0ī‚3ī€·š‘„0š‘”2(3š‘¦0š‘„20āˆ’š‘„20š‘š‘¦0+2š‘„30š‘š‘¦0+3š‘„40š‘Ÿ+6š‘„30š‘¦20āˆ’3š‘¦30š‘„0+š‘„30š‘Ÿ2āˆ’9š‘„30š‘¦0+6š‘„40š‘¦0āˆ’9š‘„20š‘¦20+2š‘¦30š‘„20āˆ’2š‘„30š‘Ÿāˆ’2š‘Ÿš‘¦30āˆ’2š‘š‘¦30+š‘2š‘¦30+š‘Ÿ2š‘¦30+š‘„20š‘š‘¦0š‘Ÿ+3š‘„0š‘Ÿš‘¦20š‘+š‘¦20š‘„0š‘’š‘+š‘š‘„20š‘¦0š‘’āˆ’š‘š‘„20š‘¦0š‘āˆ’3š‘„0š‘š‘¦20+3š‘„0š‘¦20+3š‘¦30š‘„0š‘Ÿ+3š‘¦30š‘„0š‘āˆ’6š‘„20š‘Ÿš‘¦0+2š‘„50āˆ’3š‘„40+š‘¦30+2š‘Ÿš‘¦30š‘+9š‘„30š‘Ÿš‘¦0āˆ’6š‘„0š‘Ÿš‘¦20+9š‘„20š‘¦20š‘Ÿ+5š‘„20š‘¦20š‘+š‘„30+3š‘„0š‘Ÿ2š‘¦20+3š‘„20š‘Ÿ2š‘¦0ī‚ī‚+š‘£1,3yā‹Æ,š‘¦(š‘”)=š‘¦(0)+0ī‚€āˆ’š‘’š‘„0+š‘š‘„0āˆ’š‘’š‘¦0ī‚š‘”š‘¦0+š‘„0āˆ’12ī‚€š‘¦0+š‘„0ī‚3Ɨī‚€š‘¦0š‘”2ī‚€3š‘¦0š‘’š‘„20š‘+š‘¦20š‘š‘„0š‘’+2š‘’š‘„30š‘āˆ’š‘š‘„20š‘¦0āˆ’š‘š‘„0š‘¦20āˆ’š‘2š‘„30+š‘š‘„30š‘¦0+š‘š‘„20š‘¦0š‘Ÿ+š‘š‘„0š‘¦20š‘+š‘š‘„20š‘¦20+š‘š‘„0š‘¦20š‘Ÿāˆ’š‘’2š‘„30āˆ’3š‘¦0š‘’2š‘„20āˆ’3š‘¦20š‘’2š‘„0āˆ’š‘¦30š‘’2ī‚ī‚+š‘£2,3ā‹Æ.(2.14)

3. Numerical Results and Comparison with ADM

For comparison with the results obtained by ADM [18], the parameter values in four cases are considered in Table 1.


Case š‘„ 0 š‘¦ 0 š‘ š‘ š‘’ š‘Ÿ

1 0.5 0.3 0.8 0.2 0.5 0.9
20.50.30.80.20.50.1
30.50.60.50.50.30.1
40.50.20.50.50.10.2

Results of four terms approximation for š‘„(š‘”),š‘¦(š‘”) obtained by using HPM and ADM [18] are presented in (3.1), respectively: Case1āˆ¶š‘„ā‰ˆ0.5āˆ’0.35š‘”+0.19476š‘”2āˆ’0.107288š‘”3,š‘¦ā‰ˆ0.3āˆ’0.1125š‘”+0.018808š‘”2āˆ’0.0011284š‘”3,Case2āˆ¶š‘„ā‰ˆ0.5+0.05š‘”+0.012265š‘”2āˆ’0.0016032š‘”3,š‘¦ā‰ˆ0.3āˆ’0.1125š‘”+0.024433š‘”2āˆ’0.00398199š‘”3,Case3āˆ¶š‘„ā‰ˆ0.3+0.0799t+0.00533t2āˆ’0.00115š‘”3,š‘¦ā‰ˆ0.6āˆ’0.08š‘”+0.01866š‘”2āˆ’0.00231š‘”3,Case4āˆ¶š‘„ā‰ˆ0.5+0.07857š‘”āˆ’0.016020š‘”2āˆ’0.00119873š‘”3,š‘¦ā‰ˆ0.2+0.051428š‘”+0.0055918š‘”2+0.00002245š‘”3,Case1āˆ¶š‘„ā‰ˆ0.5āˆ’0.35000š‘”+0.19476š‘”2āˆ’0.10728š‘”3,š‘¦ā‰ˆ0.3āˆ’0.11250š‘”+0.018809š‘”2āˆ’0.0011286š‘”3,Case2āˆ¶š‘„ā‰ˆ0.5+0.05000š‘”+0.012266š‘”2āˆ’0.0016034š‘”3,š‘¦ā‰ˆ0.3āˆ’0.11250š‘”+0.024434š‘”2āˆ’0.0039821š‘”3,Case3āˆ¶š‘„ā‰ˆ0.3+0.08000t+0.005333t2āˆ’0.0011555š‘”3,š‘¦ā‰ˆ0.6āˆ’0.08000š‘”+0.018667š‘”2āˆ’0.0023112š‘”3,Case4āˆ¶š‘„ā‰ˆ0.5+0.07857š‘”āˆ’0.016021š‘”2āˆ’0.0011984š‘”3,š‘¦ā‰ˆ0.2+0.051430š‘”+0.0055920š‘”2+0.00002246š‘”3.(3.1) Figures 1ā€“4 show the relations between prey and predator populations versus time.

A noteworthy observation from Figure 1 is that prey and predator species can become extinct simultaneously for some values of parameters, regardless of the initial values. Thus, overexploitation of the prey population by constant effort harvesting process together with high predator capturing rate may lead to mutual extinction as a possible outcome of predator-pray interaction. In Figure 2, only the predator population gradually decreases and becomes extinct despite the availability of increasing prey population. This can be attributed to the effect of the predator death rate, being greater than the conversion rate and low constant prey harvesting as shown in Case 2 (see Table 1). Figures 3 and 4 illustrate the possibility of predator and prey long-term coexistence. Depending on the initial values, both prey and predator populations increase or reduce in order to allow long-term coexistence [18].

4. Conclusion

Homotopy perturbation method was employed to approximate the solution of the ratio-dependent predator-prey system with constant effort prey harvesting. The results obtained here were compared with results of Adomian decomposition method. The results show that there is less computations needed in comparison to ADM.

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Copyright © 2008 Abdoul R. Ghotbi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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