Existence and Nonexistence of Traveling Wave Solutions for a Reaction – Diffusion Preys – Predator System with Switching Effect

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Introduction
Saha and Samanta [1] considered the following two preys-one predator system with switching effect: − by; where x i ðtÞ ði ¼ 1; 2Þ and yðtÞ are the densities of two preys and one predator, respectively.α 1 ; α 2 ; K 1 ; K 2 ; β 1 ; β 2 ; τ 1 ; τ 2 and b are positive constants, b is natural mortality.For more specific background details on this system, we can take a look at [1].Intrapopulation competition of the predator is a key factor in accurately predicting the population spread of the model.Moreover, due to the uneven distribution of preys and predators in different spaces, in the current paper, we study the following PDE: where m is reduction rate of intrapopulation competition, d 1 ; d 2 ; d 3 denote the diffusion coefficients, respectively, which are positive constants.If α 2 β 1 K 1 ¼ α 1 β 2 K 2 , by direct calculation, the system (2) has the planar equilibrium point and interior equilibrium point E 1 ðK 1 ; K 2 ; 0Þ and E 2 ðη * 1 ; η * 2 ; θ * Þ, and where η * 1 is a real and positive root of the equation In the past three decades, the existence and asymptotic behavior of solutions for some models had been studied by many scholars.Zhang and Ouyang [2] proved the existence of global weak solutions for a viscoelastic wave equation with memory term, nonlinear damping and source term by using the potential well method combined with Galerkin approximation procedure.Zhang and Miao [3], using Glerkin method and the multiplier technique, obtained the existence and asymptotic behavior of strong and weak solutions for nonlinear wave equation with nonlinear damped boundary conditions, respectively.
Population ecology has been well-developed as an important branch of biomathematics, in which the existence and nonexistence of traveling wave solutions of biological system, is one of the most in-depth researches by scholars, where the Lotka-Volterra model has attracted much attentions.Dunbar [4,5] in the known papers proved the existence of traveling wave solutions to a special prey-predator model by applying Lyapunov function.He proposed a two-step method for the existence of traveling wave solutions of some specific systems for prey and predator interactions.The first step, applying shooting argument, he demonstrated the existence of semitraveling wave solutions.The second step, he proved the semitraveling wave solutions actually connect to the positive equilibrium point by using the Lyapunov functions method.Lin et al. [6] studied the one prey-two predators model, and proved existence of traveling wave front connecting the trivial equilibrium point and the positive equilibrium point with some certain conditions by using the cross iteration method.Due to the variety and inhomogeneity of ecosystems, the study of the general diffusive prey-predator model has more important significance.Wang and Fu [7], by establishing Lyapunov function, proved the existence of traveling waves solutions to the reaction-diffusion prey-predator models with kinds of functional responses, may be decided by the predator and prey populations at the same time.Hsu and Lin [8] considered general diffusive prey-predator models.First, using the method of counter evidence they proved that the general diffusive predator-prey models has no positive traveling wave solutions under specific conditions.Then, applying the method of super-sub solutions, they proved existence of semitraveling wave solutions.Final, establishing the strictly contracting rectangles they concluded existence of traveling wave solutions.Huang and Ruan [9] studied the existence of traveling wave solutions for a reaction-diffusion system.Ai et al. [10] by constructing Lyapunov function and using the squeeze method proved a similar general existence result.For more results, we can see [11][12][13][14][15][16][17][18] and the references therein.
A solution ðx 1 ðμ; tÞ; x 2 ðμ; tÞ; yðμ; tÞÞ for system (2) is called a traveling wave solution when it has the special form where the wave speed c is positive constant, and ðX 1 ; X 2 ; YÞ satisfies the following ODE system: and the boundary conditions as follows: where Y 0 is a positive constant.
In this paper, based on the idea from Ai et al. [10], we consider traveling wave solutions for two preys-one predator systems (2) with switching effect.We prove that the nonexistence and existence of traveling wave solutions of system (2), namely, we show the nonexistence and existence of positive solutions of system (6) satisfying (7), (8) and (9).Let us point out that although this idea has been used by the others, our application is new.Our problem is more difficult to solve, and we need more precise calculations.
The structure of the paper is organized as follows.Section 2 is devoted to the proof of nonexistence of semitraveling wave solutions for the system (2) by using linearization method.Section 3 is concerned with existence of semitraveling wave solutions by method of the super-sub solution and Schauder fixed point theorem.Such semitraveling wave solutions connect the planar equilibrium point E 1 ðK 1 ; K 2 ; 0Þ at ξ → − 1.In Section 4, utilizing the Lyapunov function techniques, we show, with the aid of LaSalle's invariance principle, that semitraveling wave solutions of system (2) are traveling wave solutions.These traveling wave solutions connect the only positive equilibrium point E 2 ðη * 1 ; η * 2 ; θ * Þ at ξ → 1 under the additional conditions.In Section 5, the numerical experiments support the validity of our theoretical results.
Hereafter, for convenience, we shall apply i to represent the number 1; 2.

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Nonexistence of Semitraveling Wave Solutions
We pay attention to the nonexistence of semitraveling solutions for the system (2) in the section.Let Our main result is as following.
Proof.Linearizing the last equation of system (6) around ðK 1 ; K 2 ; 0Þ, we get Thus the characteristic equation of ( 11) is as follows: Suppose λ 1 and λ 2 are two eigenvalues of (12), namely For contradiction, we suppose ðX 1 ; X 2 ; YÞ is a positive solution of system (6) So the positive solutions of ( 11) are e pξ cos qξ and e pξ sin qξ, and they cannot be of the same sign as ξ near negative infinity.Since both eigenvalues have nonzero real parts, the stability of the original equation at equilibrium ðK 1 ; K 2 ; 0Þ is the same as that of the linearized equation at equilibrium ðK 1 ; K 2 ; 0Þ, yielding a contradiction.This proves Theorem 1.

Existence of Semitraveling Wave Solutions
In order to prove the existence of semitraveling wave solutions for system (2), we first give the definition super-sub solutions, then we construct a pair of super-sub solutions of system (2), and finally we prove the existence of semitraveling wave solutions for system (2) by applying method of super-sub solution and Schauder fixed point theorem.
The definition of super-sub solutions of (6) as following.
Definition 1.The functions ðX 1 ; X 2 ; Y Þ and ðX 1 ; X 2 ; Y Þ on R are called a pair of super-sub solutions of (6) if the following hold, where U i0 are positive constants, ðX 1 ; X 2 ; Y Þ; ðX 1 ; X 2 ; Y Þ on R are continuous functions.
(ii) There is a finite set B so that: (a) X i ; X i ; Y ; Y 2 C 2 ðR=BÞ.

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The following will provide the super-sub solutions required to show the existence of semitraveling wave solutions of the system (6) on c>c * and c ¼ c * , respectively.Assume where hold.
Constants ω; κ; ζ; B one by one in the following order such that the inequalities hold, where We define X i ðξÞ; X i ðξÞ; Y ðξÞ; Y ðξÞ on R as follows: Then the system (6) has a pair of super-sub solutions ðX 1 ; X 2 ; Y Þ and ðX 1 ; X 2 ; Y Þ.
Proof.Now we prove the above constants are well defined.First, we have so that ω is well defined.Since choice of ω yields that cω − d i ω 2 >0, the κ i is well defined, so κ is well defined.Due to the assumptions of ω; κ; ζ; B, we have a 0 <maxfa 1 ; a 2 g<minf0; a 3 g.According to the definitions of X i ; X i ; Y ; Y , it is clear that X i ðξÞ<X i ðξÞ and Y ðξÞ<Y ðξÞ, 8ξ 2 R, and Due to X 1 ≡ K 1 ; 8ξ 2 R, so we obtain For ξ<a 1 , since a 1 <a 3 , we have Y ðξÞ ¼ e λξ and the inequality 4 Advances in Mathematical Physics Due to definition of κ, so we have For ξ>a 1 , we obtain X 1 ðξÞ ¼ 0, and then holds.Similarly, we have For ξ<a 3 , we have Y ðξÞ ¼ e λξ , and by assumption ( 17), we infer And hence holds.

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For ξ<a 0 , we have By the definition of B, we obtain Y ðξÞ ≤ Y ðξÞ ≤ e λa 0 .Thus, for ξ<a 0 , it holds that Combining with the form of Y ðξÞ, for a 0 <0 and 0<ζ <ω, we conclude For ξ>a 0 , due to Y ðξÞ ¼ 0, so we obtain The proof is completed.
and ( 18) are satisfied.Constants ω; κ one by one in the following order such that the inequalities hold.There is a sufficiently large N 0 >0 such that for 8N ≥ N 0 , we define Then the system (6) has a pair of super-sub solutions ðX 1 ; X 2 ; Y Þ and ðX 1 ; X 2 ; Y Þ.
Proof.Similar Lemma 1, we can conclude that ω and κ are well defined.By the assumption of M 1 ; ω; κ and the a i ði ¼ 0; 1; 2; 3Þ, we have a 0 <a i <a 3 <0 ði ¼ 1; 2Þ, and N is sufficiently large.Let X 1 ; X 2 ; Y are continuous functions on R satisfying Due to X 1 ¼ K i − κe ωξ and a 1 <a 3 , so Y ðξÞ ¼ M 1 jξje λξ , one has Since derivative of ðjξje ðλ−ωÞξ Þ 0 >0 for ξ 2 ð − 1; − 1=ðλ − ωÞÞ and a 1 < − 1=ðλ − ωÞ, so that jξje ðλ−ωÞξ ≤ 1=½ðλ − ωÞe for ξ<a 1 .In combination with the constraint on κ, we get In addition, Similarly, we obtain For ξ<a 0 , we have For all large N, it readily follows that Advances in Mathematical Physics Since and Thus we obtain Since N is large enough, so it holds that Furthermore, we have For ξ<a 3 , now we check the Y ðξÞ.Since we can apply ( 17) and ( 18) to get The proof is completed.
□ For convenience, let Then the system ( 6) can be written as follows: We can easy verify that the F 1 ðX 1 ; X 2 ; YÞ; F 2 ðX 1 ; X 2 ; YÞ; GðX 1 ; X 2 ; YÞ satisfy Lipschitz condition on ½0; where Ω is a positive constant.We give the following existence result of system (63) on semitraveling wave solutions.
Proof.Define the functions b where Ω is the constant in (64).We can easily check that b F 1 ðX 1 ; X 2 ; YÞ is nondecreasing in X 1 2 ½0; K 1 for every fixed ðX 2 ; YÞ 2 ½0; can be rewritten as follows: and

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We define the map Γ ¼ ðΓ 1 ; Γ By variation-of-parameters formula we obtain that ðΦ 1 ; Φ 2 ; Ψ Þ ¼ ΓðX 1 ; X 2 ; YÞ for each ðX 1 ; X 2 ; YÞ 2 W is a bounded solution to the following system Apparently, the fixed point of Γ in W is a solution of system (63).So, we are going to prove that Γ in W has a fixed point.Inspired by [10], we define the Banach space with the exponentially weighted norm here 0<ρ<min fjλ − i jg ði ¼ 1; 2; 3Þ, and we can easily know this subset W is closed, bounded and convex in X ρ : Obviously, Γ : W → W is Lipschitz continuous, and compact on W. From the Schauder fixed point theorem, it follows that Γ has a fixed point ðX 1 ; X 2 ; YÞ in W. Next, we prove that the X 0 i ; X 00 i ; Y 0 and Y 00 are bounded.Note that for ξ 2 R This shows that X 0 i and Y 0 are bounded on R, and using the system (63), the boundedness of X 00 i and Y 00 are obtained as well.Finally, we show that the solution satisfying (8).Summarizing the above results, we obtain a solution ðX 1 ; X 2 ; YÞ for (63) satisfying X i ðξÞ ≤ X i ðξÞ ≤ X i ðξÞ and Y ðξÞ ≤ YðξÞ ≤ Y ðξÞ.Then by the definitions of X i ðξÞ; X i ðξÞ; Y ðξÞ and Y ðξÞ, we have ðX 1 ; X 2 ; YÞðξÞ→ðK 1 ; K 2 ; 0Þ as ξ → − 1.Using the expressions we know ðX 0 1 ðξÞ; X 0 2 ðξÞ; Y 0 ðξÞÞ→ð0; 0; 0Þ as ξ → − 1: Therefore, ðX 1 ; X 2 ; YÞ is a positive solution satisfying (8).
The proof of Theorem 2 is given.

Numerical Simulation
Numerical simulations are vital parts for the system (2) as they support the above theorem results.In current section, we provide the numerical simulations with the arithmetic software MATLAB.We show that the values of the parameters in Table 1 are used to draw Figure 1 (for b ¼ 0:8), Figure 2 (for b ¼ 1:1), and Figure 3 (for b ¼ 1:2).We assign the values as initial conditions x 1 ðμ; 0Þ ¼ 25; x 2 ðμ; 0Þ ¼ 100; yðμ; 0Þ ¼ 0:01, and choose a small disturbance of steady-state E 1 ð25; 100; 0Þ with only two preys to simulate a predator population invading a new resource habitat.Direct computations show that E 2 ¼ ð17:3939; 69:5756; 0:3847Þ for b ¼ 0:8, so we obtain that, from Figure 1, the system (2) admits a traveling wave solution, and the trajectory approaches connecting E 2 .The Figures 2 and 3 show that if the predator mortality rate b decreases, and b is less than a certain value b * (1:1 ≤ b * <1:2), then the three species approach toward coexistence.

Discussion
In this paper, we investigate the existence and nonexistence of traveling wave solutions for two preys-one predator system with switching effect.In order to be more practical and accurately predict the key factors of population dispersal, the spatial diffusive behavior of population and the internal competition of predator are considered.First, we use linearization method to discuss the nonexistence of semitraveling wave solutions with the wave speed c<c * , and c * is selected as the critical value (Theorem 1).Second, we apply super-sub solution method to obtain existence of semitraveling wave solutions, only connecting the planar equilibrium point E 1 ðK 1 ; K 2 ; 0Þ with c ≥ c * (Theorem 2).Moreover, utilizing method of Lyapunov function, we obtain traveling wave solutions to system (2), namely, the semitraveling wave solutions from Theorem 2 connect the only positive equilibrium point E 2 ðη * 1 ; η * 2 ; θ * Þ at infinity (Theorem 3).Finally, we provide numerical experiments to demonstrate existence results 14 Advances in Mathematical Physics of the traveling wave solutions to system (2), and we show that the three species approach toward coexistence when the predator mortality rate is less than a certain value.

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t a n c e ( μ )