Abstract
In this paper, we are concerned with traveling wave solutions for two preys–one predator system with switching effect. First, we discuss that there is no traveling wave solution for this system by using linearization method. Second, applying super-sub solution method we establish the existence of semitraveling wave solutions with the minimal speed explicitly defined. Moreover, using the method of Lyapunov function and LaSalle’s invariance principle, under certain conditions, we obtain that the semitraveling wave solutions connect the only positive equilibrium point at infinity, are actually traveling wave solutions. Finally, the numerical experiments support the validity of our theoretical results.
1. Introduction
Saha and Samanta [1] considered the following two preys–one predator system with switching effect:where and are the densities of two preys and one predator, respectively. and are positive constants, 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 is reduction rate of intrapopulation competition, denote the diffusion coefficients, respectively, which are positive constants.
If , by direct calculation, the system (2) has the planar equilibrium point and interior equilibrium point and , andwhere is a real and positive root of the equation , with
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 [11–18] and the references therein.
A solution for system (2) is called a traveling wave solution when it has the special formwhere the wave speed is positive constant, and satisfies the following ODE system:and the boundary conditions as follows:where 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 at . 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 at under the additional conditions. In Section 5, the numerical experiments support the validity of our theoretical results.
Hereafter, for convenience, we shall apply to represent the number .
2. 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.
Theorem 1. Suppose holds. For , the system (6) does not have positive solutions satisfying (8).
Proof. Linearizing the last equation of system (6) around , we getThus the characteristic equation of (11) is as follows:Suppose and are two eigenvalues of (12), namelyFor contradiction, we suppose is a positive solution of system (6) with satisfying (8). If , so that , then positive solution of (11) is unbounded as . Suppose , then and form a complex conjugate pair: , where . So the positive solutions of (11) are and , and they cannot be of the same sign as near negative infinity. Since both eigenvalues have nonzero real parts, the stability of the original equation at equilibrium is the same as that of the linearized equation at equilibrium , yielding a contradiction. This proves Theorem 1.
3. 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 and on are called a pair of super-sub solutions of (6) if the following(i) hold, where are positive constants, on are continuous functions.(ii) There is a finite set so that:(a) .(b) The limits to satisfy:(iii) For continuous functions with / satisfy:
The following will provide the super-sub solutions required to show the existence of semitraveling wave solutions of the system (6) on and , respectively.
Assumewhere , andhold.
Lemma 1. Assume that , , and (17) is satisfied. . Constants one by one in the following order such that the inequalitieshold, whereWe define on as follows:whereThen the system (6) has a pair of super-sub solutions and .
Proof. Now we prove the above constants are well defined. First, we haveso that is well defined. Since choice of yields that , the is well defined, so is well defined.
Due to the assumptions of , we have . According to the definitions of , it is clear that and , , andLet are continuous functions satisfying and .
Due to , so we obtainIf , then . Combining with , we haveFor , since , we have and the inequalityDue to definition of , so we haveFor , we obtain , and thenholds.
Similarly, we haveFor , we have , and by assumption (17), we inferFor , since , so we haveAnd henceholds.
For , we haveBy the definition of , we obtain . Thus, for , it holds thatCombining with the form of , for and , we concludeFor , due to , so we obtainThe proof is completed.
Lemma 2. Assume that , , , , and (17) and (18) are satisfied. Constants one by one in the following order such that the inequalitieshold. There is a sufficiently large such that for , we defineandThen the system (6) has a pair of super-sub solutions and .
Proof. Similar Lemma 1, we can conclude that and are well defined. By the assumption of and the , we have , and is sufficiently large.
Let are continuous functions on satisfying and .
For , combining with , we infer thatDue to and , so , one hasSince derivative of for and , so that for . In combination with the constraint on , we getIn addition,Similarly, we obtainandFor , we haveFor all large , it readily follows thatSincesothus we haveandThus we obtainSince is large enough, so it holds thatFurthermore, we haveFor , now we check the . SinceSo that . we can apply (17) and (18) to getThuscombining withit follows thatThe proof is completed.
For convenience, let
Then the system (6) can be written as follows:
We can easy verify that the satisfy Lipschitz condition on , namelywhere is a positive constant.
We give the following existence result of system (63) on semitraveling wave solutions.
Theorem 2. If (17) and (18) hold. Then the system (63) has a positive solution for every , and satisfyingand are bounded on . Moreover, the solution satisfying (8).
Proof. Define the functions and , where is the constant in (64). We can easily check that is nondecreasing in for every fixed , is nondecreasing in for every fixed , is nondecreasing in for every fixed , then (63) can be rewritten as follows:LetandWe define the map byBy variation-of-parameters formula we obtain that for each is a bounded solution to the following systemApparently, the fixed point of in is a solution of system (63). So, we are going to prove that in has a fixed point. Inspired by [10], we define the Banach spacewith the exponentially weighted normhere , and we can easily know this subset is closed, bounded and convex in
Obviously, is Lipschitz continuous, and compact on . From the Schauder fixed point theorem, it follows that has a fixed point in . Next, we prove that the and are bounded.
Note that for It follows that , and for , whereThis shows that and are bounded on , and using the system (63), the boundedness of and are obtained as well. Finally, we show that the solution satisfying (8). Summarizing the above results, we obtain a solution for (63) satisfying and . Then by the definitions of and , we have as . Using the expressionswe know as Therefore, is a positive solution satisfying (8).
The proof of Theorem 2 is given.
4. Existence of Traveling Wave Solutions
Summarizing the above results, Theorem 2 established that the system (63) has a positive solution , and . In what follows, we aim to verify the solution satisfying by applying method of Lyapunov function.
Theorem 3. Suppose that all conditions in Theorem 2 are met. Furthermore, assume that , , and hold. Then the system (63) has a positive solution satisfying (8) and (9) for every .
Proof. We konw that system (63) admits a positive solution satisfying (8) by Theorem 2. We shall show that the as . We construct a Lyapunov function as follows:in the following regionwhereWe havewhereAnd we haveSoLet , be a positive constant solution of the following equationThen we haveandwhere .
We can appeal to the comparison theorem to conclude . If there exists such that . We let be the solution of with have a solution, then can apply the comparison theorem to derive that . NoticeThis means that as , where the is a finite number greater than . It follows as for some . Contradicting with the definition of , we have for .
Similarly, we can obtain constants such that for . It shows that and the equality hold only at .
Applying LaSalle’s invariance principle, as . Theorem 3 is proved.
5. 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 ), Figure 2 (for ), and Figure 3 (for ). We assign the values as initial conditions , and choose a small disturbance of steady-state with only two preys to simulate a predator population invading a new resource habitat. Direct computations show that for , so we obtain that, from Figure 1, the system (2) admits a traveling wave solution, and the trajectory approaches connecting . The Figures 2 and 3 show that if the predator mortality rate decreases, and is less than a certain value (), then the three species approach toward coexistence.
6. 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 , and 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 with (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 at infinity (Theorem 3). Finally, we provide numerical experiments to demonstrate existence results 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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Yujuan Jiao was supported by the National Natural Science Foundation of China (no. 12361047) and the Gansu Province Natural Science Foundation (no. 23JRRA1735).