Research Article | Open Access
An Impulsive Two-Prey One-Predator System with Seasonal Effects
In recent years, the impulsive population systems have been studied by many researchers. However, seasonal effects on prey are rarely discussed. Thus, in this paper, the dynamics of the Holling-type IV two-competitive-prey one-predator system with impulsive perturbations and seasonal effects are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses (the chemical controls) for all species and the constant impulse (the biological control) for the predator at different fixed time but, the same period. In addition, the intrinsic growth rates of prey population are regarded as a periodically varying function of time due to seasonal variations. Sufficient conditions for the local and global stabilities of the two-prey-free periodic solution are established. It is proven that the system is permanent under some conditions. Moreover, sufficient conditions, under which one of the two preys is extinct and the remaining two species are permanent, are also found. Finally, numerical examples and conclusion are given.
Recently, it is of great interest to study dynamical properties for impulsive perturbations in population dynamics. Impulsive prey-predator population systems have been discussed by a number of researchers [1–8] and, what is more, there are also many literatures on simple multispecies systems consisting of a three-species food chain with impulsive perturbations [7, 9–18]. Especially, two-prey and one-predator impulsive systems are drawing notice. For examples, Song and Li  studied dynamical behavior of a Holling type II two-prey one-predator system with impulsive effect concerning biological control and chemical control strategies at fixed time. Zhang et al. [17, 18] studied a Lotka-Volterra type two-prey one-predator system with impulsive effect on the predator of a fixed moment.
It is necessary and important to consider systems with periodic ecological parameters which might be quite naturally exposed such as those due to seasonal effects of weather or food supply . Indeed, it has been studied that dynamical systems with simple dynamical behavior may display complex dynamical behavior when they have periodic perturbations[20–22]. For this reason, in this paper, we consider the intrinsic growth rates of prey population as a periodically varying function of time due to seasonal variations. The seasonality is superimposed as follows[19–22]:where the parameter represents the degree of seasonality, is the magnitude of the perturbation in , and is the angular frequency of the fluctuation caused by seasonality. It is pertinent to point out that the forced ecosystem we are studying in this paper is similar to forced nonlinear oscillators in physics such as the Duffing oscillator.
Thus, we develop the Holling-type IV two-competitive-prey one-predator system with seasonality by introducing a proportional periodic impulsive poisoning(spraying pesticide) for all species and a constant periodic releasing, or immigrating, for the predator at different fixed time as follows:where are intrinsic rates of increase, are the coefficients of intra-specific competition, are parameters representing competitive effects between two preys, are the per-capita rates of predation of the predator, are the half-saturation constants, denotes the death rate of the predator, are the rates of conversing prey into predator, are the magnitude, are the angular frequency, are the period of spaying pesticides (harvesting) and the impulsive immigration or stock of the predator, respectively, present the fraction of the preys and the predator which die due to the harvesting or pesticides, and is the size of immigration or stock of the predator.
In Section 2, we give some notations and lemmas. In Section 3, we show the boundedness of the system and take into account the local and global stabilities of two-prey-free periodic solutions by using Floquet theory for the impulsive equation, small amplitude perturbation skills and comparison techniques, and finally, prove that the system is permanent under some conditions. Moreover, we give the sufficient conditions under which one of the two prey extinct and the remaining two species are permanent. Numerical examples are given in Section 4.
Let , , and Denote the set of all of nonnegative integers and the right hand of the first three equations in (1.2). Let , then is said to belong to class if (1) is continuous on , and exists, where and ,(2) is locally Lipschitzian in .
Definition 2.1. Let . For , the upper right derivative of with respect to the impulsive differential system (1.2) is defined asThe solution of system (1.2) is a piecewise continuous function , is continuous on . Obviously, the smoothness properties of guarantee the global existence and uniqueness of solutions of system (1.2) [23, 24].Definition 2.2. The system (1.2) is permanent if there exist such that, for any solution of system (1.2) with ,We will use a comparison result of impulsive differential inequalities. Suppose that satisfies the following hypotheses.
is continuous on and the limit exists, where and , and is finite for and .Lemma 2.3 (see ). Suppose andwhere satisfies , and are nondecreasing for all . Let be the maximal solution for the impulsive Cauchy problemdefined on . Then, implies that , where is any solution of (2.3).We now indicate a special case of Lemma 2.3 which provides estimations for the solution of a system of differential inequalities. For this, we let denote the class of real piecewise continuous (real piecewise continuously differentiable) functions defined on .Lemma 2.4 (see ). Let the function satisfy the inequalitieswhere and and are constants and is a strictly increasing sequence of positive real numbers. Then, for , Similar result can be obtained when all conditions of the inequalities in the Lemmas 2.3 and 2.4 are reversed. Using Lemma 2.4, it is possible to prove that the solutions of the Cauchy problem (2.4) with strictly positive initial value remain strictly positive.Lemma 2.5. The positive octant is an invariant region for system (1.2).Proof. Let be a solution of system (1.2) with a strictly positive initial value . By Lemma 2.4, we can obtain that, for ,where , , and . Thus, and remain strictly positive on .
Now, we give the basic properties of an impulsive differential equation as follows:System (2.8) is a periodically forced linear system. It is easy to obtain that, is a positive periodic solution of (2.8). Moreover, we can obtain thatis a solution of (2.8). From (2.9) and (2.10), we get easily the following result.Lemma 2.6. for all solutions of (2.8) with .Therefore, system (1.2) has a two-prey-free periodic solution
3. Main Results
Theorem 3.1. The periodic solution of system (1.2)
is globally asymptotically stable if for where .Proof. First,
we will prove the local stability of the periodic solution . For this, consider the following impulsive
differential equation:Then, , , and by Lemma 2.3,
where and are solutions
of systems (1.2) and (3.2), respectively. Thus we will show the local stability
of the solution of system
(3.2), where . The local stability of the two-pest-free periodic
solution may be
determined by considering the behavior of small amplitude perturbations of the
solution. Let be any solution
of system (3.2). Define . Then, they may be written aswhere satisfiesand , the identity matrix. So the fundamental solution
matrix isThe resetting impulsive
conditions of system (3.2) becomeNote that all eigenvalues
ofare , , and . Sincewe obtain from (3.1) that the
conditions and hold.
Therefore, from the Floquet theory , we obtain is locally
Now, to prove the global stability of the two-prey-free periodic solution, let be a solution of system (1.2). From (3.1), we can take a sufficiently small number satisfyingIt follows from the first equation in (1.2) that for and for . Then, from Lemma 2.3, we have , where is a solution of the following impulsive differential equation:Since as , for any with large enough. For simplicity, we may assume that for all . Similarly, we get for any and . Since for , we can obtain from Lemmas 2.3 and 2.6 thatfor sufficiently large. Without loss of generality, we may suppose that (3.11) holds for all . From (1.2), and (3.11) we obtainIntegrating (3.12) on , we getand thus which implies that as . Further, we obtain, for ,which implies that as . Similarly, we obtain as . Now, take sufficiently small positive numbers and satisfying to prove that as . Without loss of generality, we may assume that and for all . It follows from the third equation in (1.2) that, for and ,Thus, by Lemma 2.3, we induce that , where is the solution of (2.8) with changed into . Therefore, by taking sufficiently small and , we obtain from Lemma 2.6 and (3.11) that tends to as .Let for . Then , , and . Thus has a unique positive solution .Corollary 3.2. The periodic solution of system (1.2) is globally asymptotically stable if .From the proof of Theorem 3.1, we can easily get the following corollary.Corollary 3.3. The periodic solution of system (1.2) is locally stable if Furthermore, the periodic solution of system (1.2) may remain globally stable even if there are no the seasonal effects on system (1.2).Corollary 3.4. Suppose that. Then, the periodic solution of system (1.2) is globally asymptotically stable ifNow, we show that all solutions of system (1.2) are uniformly bounded.Theorem 3.5. There is an such that , and for all large enough, where is a solution of system (1.2).Proof. Let be a solution of (1.2) with and let for . Then, if and , we obtain that . From choosing , we have, for and ,As the right-hand side of (3.18) is bounded from above by , it follows thatIf , then and if , then , where . From Lemma 2.4, we get thatwhere . Since the limit of the right-hand side of (3.20) as isit easily follows that is bounded for sufficiently large . Therefore, and are bounded by a constant for sufficiently large .Theorem 3.6. System (1.2) is permanent if ,where
Proof. Let be a solution
of system (1.2) with From Theorem
3.5, we may assume that and . Thus, we only need to prove the existence of the
lower bound . For this, we consider the following impulsive differential
equation:Then, , and by Lemma 2.3,
where , and are solutions
of systems (1.2) and (3.24), respectively. So, we will show that , and . As in the proof of Theorem 3.1, we can show that and for . Let for . Since for , it follows from Lemmas 2.3 and 2.6 that and hence for
sufficiently large . Thus we only need to find and such that and for large enough.
We will do this in the following two steps.
Step 1. First, take sufficiently small positive numbers and such that , and . We will prove, there exist such that and . Suppose not. Then that, we have only the following three cases:
(i)there exists a such that , but , for all ;(ii)there exists a such that , but , for all ;(iii) and for all .
Case (i): from (3.22) we can take small enough such that We obtain from the condition of case (i) that for , where . Thus we have and as , where is a solution of systemTherefore, we can take a such that for . Thus we getfor . Let be such that . Integrating (3.17) on , we can obtain that . Thus as , which is a contradiction to the boundedness of .
Case (ii): the same argument as the case (i) can be applied. So we omit it.
Case (iii): we choose sufficiently small so that Then we obtain for where It follows from Lemmas 2.3 and 2.6 that and as , where is a solution of the following system:Thus there exists a such that for andfor . Let be such that . Integrating (3.30) on , we can obtain that . Similarly, we have as , which is a contradiction to the boundedness of . Therefore, there exist and such that and .
Step 2. If for all , then we are done. If not, we may let . Then, for and, by the continuity of , we have . In this step, we have only to consider two possible cases.
(i) Suppose that for some . Then, Select such that and , where . Let . In this case, we will show that there exists such that . Otherwise, by (2.10) and (3.26) with , we have