On the dynamics of one-prey-n-predator impulsive reaction-diffusion predator–prey system with ratio-dependent functional response

ABSTRACT In this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator–prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator–prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end.


Introduction
Reaction-diffusion equations can be used to model the spatiotemporal distribution and abundance of organisms. A typical form of reaction-diffusion population model is where u(x, t) is the population density at a space point x and time t, D > 0 is the diffusion constant, u is the Laplacian of u with respect to the variable x, and f (x, u) is the growth rate per capita, which is affected by the heterogeneous environment. Such an ecological model was first considered by Skellam [14]. Similar reaction-diffusion biological models were also studied by Fisher [5] and Kolmogoroff et al. [7] earlier. In the past two decades, the reaction-diffusion models, especially in population dynamics, have been studied extensively. For example, Ainseba and Aniţa in [1] considered a 2 × 2 system of semilinear partial differential equations of parabolic-type to describe the interactions between a prey population and a predator population and obtained some necessary and sufficient conditions for stabilizability. Xu and Ma in [21] studied a reaction-diffusion predator-prey system with non-local delay and Neumann boundary conditions and established some sufficient conditions on the global stability of the positive steady state and the semitrivial steady state. Shi and Li in [12] presented a diffusive Leslie-Gower predator-prey system with ratio-dependent Holling type III functional response under homogeneous Neumann boundary conditions. They investigated the uniform persistence of the solutions semi-flows, the existence of global attractors, local and global asymptotic stability of the positive constant steady state of the reaction-diffusion model by using comparison principle, the linearization method and the Lyapunov functional method, respectively. The results showed that the prey and predator would be spatially homogeneously distributed as time converges to infinities. Yu, Deng and Wu in [22] discussed the semi-implicit schemes for the non-linear predator-prey reaction-diffusion model with the space-time fractional derivatives, they theoretically proved that the numerical schemes are stable and convergent without the restriction on the ratio of space and time step-sizes and numerically further confirmed that the schemes have first order convergence in time and second order convergence in space. Moreover, they obtained the results that the numerical solutions preserve the positivity and boundedness. More articles on the reaction-diffusion population dynamics, please see [4,6,13,17,20].
There are many examples of evolutionary systems which at certain instants are subjected to rapid changes. In the simulations of such processes, it is frequently convenient and valid to neglect the durations of rapid changes. The perturbations are often treated continuously. In fact, the ecological systems are often affected by environmental changes and other human activities. These perturbations bring sudden changes to the system. Systems with such sudden perturbations referring to impulsive differential equations have attracted the interest of many researchers in the past 20 years since they provided a natural description of several real processes. Process of this type is often investigated in various fields of science and technology, physics, population dynamics [3,19,23], epidemics [24], ecology, biology, optimal control [8] and so on.
Recently, some impulsive reaction-diffusion predator-prey models have been investigated. Especially, Akhmet et al. [2] presented an impulsive ratio-dependent predator-prey system with diffusion; meanwhile, they obtained some conditions for the permanence of the predator-prey system and for the existence of a unique globally stable periodic solution. Wang et al. [18] generalized the above impulsive ratio-dependent system to n+1 species and got some analogous results. It is worth noting that the two models mentioned above did not involve the intra-specific competition of the predators. However, it should be concerned in most predator-prey systems, especially in the environment where food are abundant.
Motivated by the above works, we present and study the following one-prey-npredator impulsive reaction-diffusion predator-prey system with ratio-dependent functional response in this paper: In this system, it is assumed that the predator and prey species are confined to a fixed bounded space domain ⊂ R n with smooth boundary ∂ and are non-uniformly distributed in the domain. Furthermore, they are subjected to short-term external influence at fixed moment of time t k , where {t k }, k = 1, 2, . . . is a sequence of real numbers 0 = t 0 < t 1 < · · · < t k < · · · with lim k→∞ t k = +∞. Denote by ∂/∂n the outward derivative, = ∪ ∂ , and u = ∂ 2 u/∂x 2 1 + · · · + ∂ 2 u/∂x 2 n the Laplace operator. In Equations (1)-(3), D 0 u 0 and D d u d (d = 1, 2, . . . , n) reflect the non-homogeneous dispersion of population. The coefficient D s (s = 0, 1, 2, . . . , n) is the diffusion coefficient of the corresponding species. It is a measure of how efficiently the animals disperse from a high to a low density. The Neumann boundary conditions (5) characterize the absence of migration. In the absence of predators, the prey species has a logistic growth rate. We assume that the predator functional response has the form of the ratio-dependent functional response function In this paper, we will investigate the asymptotic behaviour of non-negative solutions for impulsive reaction-diffusion system (1)- (5). Note that according to biological interpretation of the solutions u s (t, x) (s = 0, 1, . . . , n), they must be non-negative. We will give conditions for the long-term survival of each species in terms of permanence. The permanence of the system indicates that the number of individuals of each species stabilizes on certain boundaries with respect to time.
This paper is organized as follows. In Section 2, we give some basic assumptions and useful auxiliary results. Conditions for the ultimate boundedness of solutions and permanence of the system are obtained in Section 3. In Section 4, we establish conditions for the existence of the unique periodic solution of the system. Examples and numerical simulations are presented in Section 5 to verify the feasibility of the results. Finally, we discuss the obtained results and present some interesting problems.

Preliminaries
Let N and R be the sets of all positive integers and real numbers, respectively, and R + = [0, ∞). The following assumptions will be needed throughout the paper. Conditions of periodicity are natural because of the seasonal changes and biological rhythms.
Consider the following impulsive logistic differential equation: where z ∈ R + , a and b are positive constants, strictly increasing sequence {t k } satisfies where z 0 is given below. Then we have the following useful result.
Proof: For t ∈ [0, t 1 ], we have that It is obvious that the solution is positive-valued and no larger than max{z 0 , b} on the interval. Moreover, if θ t t 1 , then Particularly, 0 < z(t 1 ) Q 1 , hence, 0 < z(t + 1 ) = z(t 1 )λ 1 (z(t 1 )) Q 2 . It is easy to show that 0 < z(t) max{Q 1 , . Similarly to (8), we can verify that 0 < z(t 2 ) Q 1 . Furthermore, using the same analysis, we can show that 0 < z(t) Q 3 if t ∈ (t k , t k+1 ], k = 2, 3, . . .. Now, let us give another useful lemma. Consider the following vector impulsive differential equation where w = (w 0 , w 1 , . . . , w n ) ∈ L n+1 p ( ), p > n is a positive integer. The operator A has the domain D(A) = {ξ : ξ ∈ W 2,p ( ), (∂ξ /∂n)| ∂ = 0}, where W 2,p ( ) is the Sobolev space of functions defined as in whichf has the norm where is the Gamma function. The operator A −α is bounded and bijective. The operator Here, · is the norm in the space X = L n+1 p . We denote by C m+α ( ), where m is a positive integer and 0 < α < 1, the space of mtimes continuously differentiable functions f : → R, which have m-order derivatives satisfying the Hölder condition with exponent α.
By Theorem 9 of paper [2], we have the following lemma immediately.
Then the set The following lemmas will be needed throughout the paper.

and u(t, x) satisfies the following inequalities:
On the basis of the upper and lower solution method for quasi-monotone systems (see [10]), we can verify that, for continuously differentiable initial functions u s0 (x) :¯ → R + , as well as u s0 (x) are not identically zero for all s = 0, 1, . . . , n, there exists a classical solution of system (1)- (3) and (5), which can be extended to the semi-axis t > 0. A vector-function is the classical solution of system without impulses (1)- (3) and (5), if it is of class C 2 in x, x ∈ , of class C 1 in x, x ∈¯ , of class C 1 in t, t > 0, and satisfies the system.
According to biological interpretation, we only consider the non-negative solutions of the system. Hence, the following assertion is of major importance.

Proof: For Equation
are its lower and upper solutions. Then, since u 00 (x) 0 and u 00 (x) is not identically zero, by Lemma 2.4, we getû 0 (t, . Taking into account positiveness of the function f 01 , we can repeat the same argument to prove the positiveness of For Equations (2) and (3), it can be also verified thatû are their lower and upper solutions, respectively. Using the same analysis, we finally have . . , n and t ∈ (0, ∞).

Permanence
In this section, applying the upper and lower solution method and comparison theory of differential equations, we establish some sufficient conditions for the ultimate boundedness and permanence of the system. Before this, two definitions are given firstly.
Through the above analysis, we get some conditions under which the two species are permanent. Then, we will give some conditions that will lead to extinction of the predator species.
From the inequalities applying the comparison theorem, we can find that u i (t, x, u 00 , u 10 , . . . , u n0 ) ū i (t, M u i ) for t t 1 . Moreover, using impulsive condition (5), we obtain that Proceeding in this fashion, we conclude that solutions of Equations (2) with impulses are bounded from above by the corresponding solutions of linear impulsive equations Taking into account (21), we see that all solutions of the last equation tend to zero as t → ∞.

Periodic solutions
In the following, we study the existence of the periodic solution by constructing an appropriate auxiliary function. We will note that the conditions of the existence of the periodic solution are dependent on the permanence of the system. ln and other elements of the matrix are equal to zero. Then system (1)-(5) has a unique globally asymptotically stable strictly positive piecewise continuous τ -periodic solution.

Remark 4.1:
This paper generalizes the models investigated in [2] and [18] by adding intra-specific competition terms of predators. If we do not consider these effects (i.e. m = 0), and take β s (t, x) = 1 for all (t, x) ∈ R ×¯ and all s = 0, 1, . . . , n, the model presented in this paper will degrade into that introduced in [18]. Comparing the corresponding results such as ultimate boundedness, permanence and periodic solutions between them, we will find that the present paper owns the same sufficient conditions with paper [18]. Moreover, if we take m = 0, β s = 1 and n = 1, i.e. there is only one predator and no intra-specific competition, the present model will degrade into that studied in [2]. Also, this paper admits the same conditions with paper [2] on the corresponding theorems.

Conclusion and discussion
In this paper, we present and study a one-prey-n-predator impulsive reaction-diffusion periodic predator-prey system with ratio-dependent functional response. The reactiondiffusion term shows that our prey and predator species are only confined in an isolated habitat for which the impact of migration, including both emigration and immigration, is presumably negligible, such as a remote patchy forest or an isolated island or a lake ecosystem which is practically water islands with distinct boundaries. Some sufficient conditions for the permanence (Theorems 3.1 and 3.2), extinction (Theorem 3.3) and the existence of a unique globally stable positive periodic solution (Theorem 4.1) of the system are established. By Theorem 3.1, we see that the prey species with intra-specific competition is ultimately bounded if the impulsive coefficients are bounded for any bounded solution of the system, so do the predator species which have the intra-specific competitions (Equation (2)). For each predator species without intra-specific competition (Equation (3)), its ultimate boundedness requires a negative average growth rate in a τperiod in the absent of the prey species by Theorem 3.1. By Theorem 3.2, we see that if the prey or predator species want to be permanent, a positive average growth rate in a τ -period must be satisfied. However, under the permanence of the prey species, the predator species are still extinct if their average growth rates in a τ -period are negative, which are shown by Theorem 3.3. We note that if the system is permanent, by Theorem 4.1, it has a unique globally asymptotically stable strictly positive piecewise continuous τ -periodic solution if p l=1 ln K l + τ λ M < 0. Thus, there is an interesting problem. If this system has a unique globally asymptotically stable strictly positive τ -periodic solution, the system should be permanent. It seems that Theorem 4.1 may imply Theorem 3.2. But unfortunately, we cannot claim it is true, since the condition p l=1 ln K l + τ λ M < 0 in Theorem 4.1 is dependent of σ and N, which are determined by the bound of solutions when the system is permanent. Whether is there a unique globally asymptotically stable τ -periodic solution only if conditions (16), (17) and (18) hold (the system is permanent)? Can we find other conditions independent of σ and N that can guarantee the existence of the unique globally stable τ -periodic solution? We will continue to study these problems in the future.
In this paper, we only studied the system with ratio-dependent functional response, whether other type functional responses such as Holling type II, Holling type III and Beddington-DeAnglis functional response can be discussed with the same methods or not, still remain open problems.

Disclosure statement
No potential conflict of interest was reported by the authors.