CMMM Computational and Mathematical Methods in Medicine 1748-6718 1748-670X Hindawi 10.1155/2019/4849393 4849393 Research Article Analytical Solutions of a Modified Predator-Prey Model through a New Ecological Interaction https://orcid.org/0000-0001-6227-5817 Aljahdaly Noufe H. 1 Alqudah Manar A. 2 Penedo Manuel F. G. 1 Department of Mathematics Faculty of Sciences and Arts-Rabigh Campus King Abdulaziz University Jeddah Saudi Arabia kau.edu.sa 2 Mathematical Science Department Princess Nourah Bint Abdulrahman University P.O. Box 84428 Riyadh 11671 Saudi Arabia pnu.edu.sa 2019 16102019 2019 22 05 2019 20 08 2019 17 09 2019 16102019 2019 Copyright © 2019 Noufe H. Aljahdaly and Manar A. Alqudah. 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.

The predator-prey model is a common tool that researchers develop continuously to predict the dynamics of the animal population within a certain phenomenon. Due to the sexual interaction of the predator in the mating period, the males and females feed together on one or more preys. This scenario describes the ecological interaction between two predators and one prey. In this study, the nonlinear diffusive predator-prey model is presented where this type of interaction is accounted for. The influence of this interaction on the population of predators and preys is predicted through analytical solutions of the dynamical system. The solutions are obtained by using two reliable and simple methods and are presented in terms of hyperbolic functions. In addition, the biological relevance of the solutions is discussed.

Princess Nourah Bint Abdulrahman University
1. Introduction

Many phenomena in natural science, biology, physics, or engineering are studied by developing a mathematical model that consists of partial differential equations. A prey-predator system is a well-known mathematical model for studying the species’ population and density [1, 2]. Many different interactions in this model are very significant phenomena in nature’s population [3, 4]. Alqudah  modified the diffusive predator-prey model in which two predators interact with one or more preys in the mating period as follows:(1)utD1Δu=a1ua2u2a3uva4uv2,vtD2Δv=a3uva5va6v2+a4uv2.

The biological meaning of each term is presented in Table 1. We assume that ai for all i=1,2,,6 are positive constants and D=D1/D2, D1=0,D20 which means that the predator is moving toward the prey.

Description of variables and parameters of model (1).

Variable Description
u = u x , t Density of prey
v = v x , t Density of predator
D 1 Δ u Diffusion term of prey species
D 2 Δ v Diffusion term of predator species
D 1 and D2 Diffusion coefficients which are small positive constants
a 1 Growth rate of the prey
a 2 Decay rate of prey due to competition on the food supply between the male and the female
a 3 Decay rate of prey due to the interaction between one predator and one prey
a 4 Decay rate of prey due to the interaction between two predators and one prey
a 5 Mortality rate of predator
a 6 Decay rate of predator due to competition on the food supply between the male and the female

This model focuses on the term uv2 which refers to a positive interaction with respect to the predator species, like sexual interaction in the mating period, to proliferate the predators. Thus, the males and females are together and feed on the same prey or more, but this term is negative with respect to the prey. In , the existence and the uniqueness of the solution of model (1) are studied, and the scaling is introduced to get the dimensionless parameters. Thus, model (1) can be expressed as(2)Ut=UU2UVUV2,Vt=VXXk2k1Vk1V2+k1VU+k1UV2,where k1=a4a1/a2 and k2=a5/a1k1.

One way to understand the applications and dynamic system is finding the analytical or approximate solutions. In the field of mathematical computation, several methods have been developed to find the analytical solutions such as the tanh method , the Riccati equation expansion method , the (G/G2)-expansion method [8, 9], the algebraic method , and so on. In this paper, the exact and stable solutions of the system are obtained using two sufficient methods: (i) the (G/G)-expansion method  and (ii) the generalized auxiliary equation method .

This paper is organized as follows. Section 2 is the description of two employed methods. Section 3 presents the analytical solutions of the considered dynamical system. Section 4 is the discussion of the results, including the biological interpretation. The last section is the conclusion of the work and the results.

2. Description of Algorithms

This section shows briefly the steps of applying the function-expansion method for finding analytical solutions. We chose two methods that are simple and reliable using the asymptotic software (MATHEMATICA), which are the (G/G)-expansion method and the generalized auxiliary equation method. Let us consider the nonlinear evaluation equation of the form(3)P1U,V,Ut,Vt,Ux,Vx,Uxx,Vxx,=0,P2U,V,Ut,Vt,Ux,Vx,Uxx,Vxx,=0,where P1 and P2 are polynomials in U,V and their derivatives and Ux,t and Vx,t are two unknown functions. The methods that are considered in this article have the common following steps:

Transforming the function Ux,t and Vx,t to u1ξ and v1ξ, respectively, by applying the transformation ξ=xct, where c is the wave speed. Thus, the system of equations (3) is reduced to the ordinary differential equation (ODE)

(4)P1u1,v1,cu1ξ,cv1ξ,u1ξ,v1ξ,u1ξξ,v1ξξ,=0,P2u1,v1,cu1ξ,cv1ξ,u1ξ,v1ξ,u1ξξ,v1ξξ,=0.

Assuming the function u1ξ and v1ξ are expressed by the following polynomials:

(5)u1ξ=i=0nαiFξi,v1ξ=i=0mβiFξi,

where n>0 and m>0 are the degrees of the polynomial of u1 and v1, respectively. We obtain the value of n and m by the homogeneous balance theory .

Substituting the polynomials (5) into equation (4) to obtain polynomials of the function Fξ.

Equating the confections of Fξi,i=1,2,,n (or m) to zero to obtain a system in terms of variables αi,βi and c.

Solving the system to find the value of αi,βi and c.

Substituting the obtained variables αi,βi and the value of the function Fξ into equation (5) to construct the solutions of U and V.

The main difference between the methods utilized in this paper is the value of the function Fξ.

The (G/G)-expansion method where F=G/G satisfies

(6)F=F2λFμ,

and G satisfies the second order ODE:

(7)G+λG+μG=0.

The function Fξ is the general solution of the equation (6) as follows:

(8)Fξ=λ24μ2A1cosh1/2λ24μξ+A2sinh1/2λ24μξA1sinh1/2λ24μξ+A2cosh1/2λ24μξλ2,for λ24μ>0,λ2+4μ2A1cos1/2λ2+4μξA2sin1/2λ2+4μξA1sin1/2λ2+4μξ+A2cos1/2λ2+4μξλ2,for λ24μ<0,A2A1+A2ξλ2,for λ24μ=0.

The generalized auxiliary equation method where F satisfies the auxiliary equation

(9)F=h0+h1F+h2F2+h3F3+h4F4.

The general solutions of the auxiliary equation (9) have several types of solutions depending on the value of hi, i=1,2,,4. The following are some types of the solution and the reader is referred to [10, 12] to find more cases.

If h0=h1=0, the solution of equation (9) is expressed as follows:

(10)Fξ=2h2sechh2ξδ1h4sechh2ξ,h2>0,δ1>0,h3=h4,2h2cschh2ξδ1h4cschh2ξ,h2>0,δ1<0,h3=h4,h2h41±tanh2/2ξ,h2>0,δ1=0,h2sech21/2h2x2h2h4tanh1/2h2xh3,h2>0,h2sec21/2h2x2h2h4tan1/2h2xh3,h2<0.

where δ1=h324h2h4.

If h0=h1=h3=0,

(11) F ξ = h 2 / h 4 sec  h h 2 x , h 2 > 0 , h 4 < 0 , h 2 / h 4 sec h 2 x , h 2 < 0 , h 4 > 0 , 1 h 4 x , h 2 = 0 , h 4 > 0.

If h0=h1=h4=0,

(12) F ξ = h 2 / h 3 sec h 2 h 2 ξ 2 , h 2 > 0 , h 2 / h 3 sec h 2 ξ 2 , h 2 < 0 , 1 h 3 ξ 2 , h 2 = 0.

3. Analytical Solutions of the Diffusive Predator-Prey System

In this section, the analytical solutions of model (1) will be found by the methods that are described in the previous section. First, introducing the traveling wave solution ξ=XcT reduces model (1) as(13a)cu1=u1u12u1v1u1v12,(13b)cv1=v1k1k2v1k1v12+k1u1v1+k1u1v12,where UX,T=u1ξ, VX,T=v1ξ, and c is the wave speed. Then, the equations (13a) and (13b) are combined as follows:(14)cu1+v1=v1k1k2v1k1v12+u1u121k1u1v11k1u1v12.

Following the steps of the algorithm in Section 2, the solutions of u1 and v1 can be expressed as follows:(15)u1ξ=α1Fξ,v1ξ=β1Fξ.

Substituting equation (15) into equation (14) to obtain(16)cα1+β1Fx+β1Fx+α1β1k1k2Fx+α1β1k11β12k1α12F2x+α1β12k11F3x=0.

3.1. The (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M77"><mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:mi>G</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>)-Expansion Method

In order to utilize the (G/G)-expansion method, the definition of F equation (6) is applied to equation (16), and then the following algebraic system is obtained:(17)α11cλ+β1cλ+λ2k1k2+2μ=0,α12α1cβ1k11β1c3λ+β1k1=0,β1α1β1k11+2=0,μα1c+β1cλ=0.

Hence, two cases of the value of μ,λ,c and α1 are realized:

Case I

(18)c=β14k112k1+2β12k112+42β1k11β12k112,μ=β12k11k1k2+22β12k11,λ=β14k112k1+2β12k112+42β13k112,α1=2β1k11.

Case II

(19)μ=0,c=1β12/β1k1114β12k14β12k112+3k1β12k1133β12k11334β1β14k112k1+2β12k112+428β14k113β12k11k1k2+2β16k114+3k12k11+322k12,λ=182β1k1+4k1β1k11+8k1β13k1134β1k118β13k1132β14k112k1+2β12k112+428β14k113β12k11k1k2+2β16k114,α1=2β1k11.

The solutions within both cases exist if k11. As we see in Figure 1, λ24μ>0 for k11. Therefore, the solution of the system is in terms of the hyperbolic function and parameters k1,β1 and k2. Assuming 0<k1<1 yields a4a1<a2 which is not a reasonable case in actual situations. Therefore, the solution is considered only for k1>1, (a4a1>a2). The following is the solution by the (G/G)-expansion method which is a kink soliton solution for U and V as we see in Figure 2:(20)Uξ=2β1k11λ24μ2A1cos h1/2λ24μξ+A2sin h1/2λ24μξA1sin h1/2λ24μξ+A2cos h1/2λ24μξλ2,Vξ=β1λ24μ2A1cos h1/2λ24μξ+A2sin h1/2λ24μξA1sin h1/2λ24μξ+A2cos h1/2λ24μξλ2.

Plot λ24μ2 versus k1 of case (I) where β1 and k2 are fixed. (a) 0<k1<,0<a4a1<a2. (b) k1>1,a4a1>a2.

Plot of the scaled density of the prey for t=x=4,4, μ=0, λ=2β12+2/3β1, α1=β1, k12/β12+1, and k2=4β14+7β12+16/9β12+18, where β1=0.5.

3.2. The Generalized Auxiliary Equation Method

This subsection presents the solution of the considered system by applying the auxiliary equation method. Thus, we use the definition of F (9) into equation (16) which yields to the following algebraic systems:(21)α1+β1=0,α1+β1h2k1k2=0,α12+12β13h32β1k1+α1β1k11=0,β12h4+α1β1k11=0,β1h12=0.

The solutions of the algebraic system aforementioned give h1=0,h2=1+k1k2,h3=4k1β1/3,h4=1/2k11β12, and h0,k1,β1 and k2 are arbitrary constants. Hence, the appropriate solution of the auxiliary equation is in terms of the hyperbolic function. Assuming h0=0 yields the solution of the problem in the following expression: (22)Ux,t=β1k1k2+1sech21/2k1k2+1xct2β12k11k1k2+1tan h1/2k1k2+1xct4β1k1/3,Vx,t=Ux,t.

The computed solution by the generalized auxiliary equation method exists if k11 and is a soliton solution in terms of k1,k2 and β1 (see Figure 3).

Plot of the scaled density of the prey equations (22) for t=x=4,4, k2=0.5,β1=1,c=1, and k1=3.

4. Discussion 4.1. Biological Implication

The solution is obtained by the (G/G)-expansion method for λ>2μ and with the condition a4>a22/a1. Thus, this solution is obtained when the decay rate of the prey population is greater due to the interaction between two predators and one prey (a4) than the decay rate of the prey population due to the competition on the food supply over the growth rate of the prey (a22/a1). The solution in Figure 2 shows the predator population increases because of plentiful prey and sexual interaction while the prey population decline because of its high consumption during the mating period of the predator in a close environmental area. Ultimately, the predator population will become dominant in the area. This situation is expected when the predators overgraze in the mating period where the prey is plentiful.

Figure 3 presents the solution by applying the generalized auxiliary equation method for k11. This solution depicts that in certain environmental area, the prey population u will grow due to the absence of the predator v. In the mating period, the predators graze in where the prey density (u) is large and as a result u will decay. However, the obtained solutions by both considered methods remain constant away from places of grazing predator during the predator-mating period and over time.

4.2. Connection with the Previous Studies

The diffusive predator-prey model has been solved numerically and analytically in some papers. The analytical solutions of the model when both species having the same diffusivities and the prey density being of the Allee type are computed by the (G/G)-expansion method , and the results are either single structure or periodic structure. Also, it was computed by the improved Riccati equation mapping method , and the solutions are presented in three structures: single, periodic, and kink. The same model was solved analytically by utilizing the exp-expansion method , and the obtained solutions were kink solutions, singular kink solutions, dark soliton solutions, bright soliton solutions, soliton solutions, singular soliton solutions, multiple soliton-like solutions, and triangular periodic solutions, but all these solutions were not discussed in terms of the actual situation.

The numerical solution of a system of nonlinear Volterra differential equations governing on the problem of prey and predator was solved numerically by the Adomian decomposition method , Runge–Kutta–Fehlberg method and Laplace Adomian decomposition method , and the results state that the number of predator increases, as the number of prey decreases.

The present paper studied the predator-prey model with adding the term uv2 and in absence of diffusion among preys D1=0. We obtained that predator’s population increases, as the prey’s population decreases in a certain interval, and the solutions are constant out of this interval. Comparing the results obtained in the present study with different modifications of the diffusive predator-prey model, it can be concluded that obtained results are new and different.

5. Conclusion

In this paper, both the (G/G)-expansion method and the generalized auxiliary equation method have been used to find new stable analytical solutions of the modified nonlinear diffusive predator-prey model by considering the term uv2. To the best of the researchers’ knowledge, this modified model has not been solved analytically or numerically in previous research. The result showed the effect of the term uv2 in the diffusive predator-prey dynamic system which leads to a decline in the prey population and an increase in the predator population in the mating period at predator’s density points. However, the idea of this modified system can be used in epidemiological researches to construct several models. For example, if we consider u as the infected human body by any disease, in many cases we use two types of medicine which is considered as v2. Thus, the interaction term uv2 refers to combatting the epidemic or the disease through two types of medicine. In addition, the prey-predator system has been studied widely in the literature regarding the time delay . In our future work, we will study time delay for our new modified system with the effect of fluctuation of the biological parameters or fuzzy parameters.

Data Availability

The data supporting this research are from previously reported studies, which have been cited.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

Noufe Aljahdaly implemented the analytical solutions with software and analyzed the data. Manar Alqudah proposed the mathematical model of this paper. All authors wrote this paper, read, and approved the final manuscript.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.

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