Dynamic Analysis of a Heterogeneous Diffusive Prey-Predator System in Time-Periodic Environment

In this paper, a heterogeneous diffusive prey-predator system is first proposed and then studied analytically and numerically. Some sufficient conditions are derived, including permanence and extinction of system and the boundedness of the solution. The existence of periodic solution and its stability are discussed as well. Furthermore, numerical results indicate that both the spatial heterogeneity and the time-periodic environment can influence the permanence and extinction of the system directly. Our numerical results are consistent with the analytical analysis.


Introduction
Due to the complexity of ecosystems, prey-predator dynamics have always drawn interest among mathematical ecologists, as well as experimental ecologists [1][2][3]. e significance of studying prey-predator dynamics is to gain insights into the complex ecological processes. Prey-predator models, as the base of researching prey-predator dynamics, have attracted increasing attention [4][5][6][7]. Since Holling [8] introduced the concept of the functional response, a lot of studies have been devoted to the understanding of the effect of functional response on preypredator dynamics [9]. Usually, the functional response is assumed to be either prey dependent or ratio dependent in prey-predator models [10,11].
A classical general prey-predator system can be written as follows [12]: where N and P denote the prey and predator densities, respectively, f(N) is the prey growth rate, g(N, P) is the functional response, and h(g(N, P), P) is the per capita growth rate of predators. Let h(g(N, P), P) � eg(N, P) − m(P), then equation (1b) can be rewritten as follows: dP dt � (eg(N, P) − m(P))P, (2) where e is the conversion efficiency and m(P) is the specific mortality of predators in absence of prey. For the function m(P), the most widely accepted assumption [13] is m(P) � μ, where μ is a constant describing the death rate of the predator. However, Cavani and Farkas [14] introduced another function for m(P): where c is the mortality at low density and δ is the limiting, maximal mortality (obviously, c < δ). e specific mortality (3) depends on the quantity of predators, which suggests that the predator mortality is neither a constant nor an unbounded function, and increasing with quantity. Obviously, when c � δ, equation (3) can be simplified to a constant death rate type. Prey-predator systems with this nonconstant death rate have been studied by some researchers [15][16][17]. Additionally, in order to understand patterns and the mechanisms of spatial distribution of interacting species, the dispersal process is taken into consideration [18][19][20]. us, the spatiotemporal dynamics of a prey-predator system can be presented by a couple of reaction-diffusion equations based on equations (1a) and (2) [10,21,22]: zP dt � (eg(N, P) − m(P))P + D P ΔP, where D N and D P are the prey and predator diffusion coefficients, respectively, and the Laplace operator Δ describes the spatial dispersal.
Because of the emergence of Lotka-Volterra models [23,24], a logistic type growth f(N) is usually assumed for the prey species in the models. Some functional response g(N, P) are taken into account in many works, such as Holling type [25], Michaelis-Menten type [26,27], and Beddington-DeAngelis type [28,29]. Especially, many biologists argued that the ratio-dependent theory is more suitable for describing prey-predator systems in many situations [13,[30][31][32]. Since Ardini and Ginzburg proposed the ratio-dependent prey-predator system, the prey-predator systems with ratio-dependent functional response are widely studied [13,[33][34][35][36], and many interesting results are obtained.
Based on model (4a) and (4b), in this paper, we employ the ratio-dependent functional response and the nonconstant death rate (i.e., equation (3)) and assume that the growth rate of prey population follows the logistic growth type. Moreover, let u and v be the prey density and the predator density, respectively. en, the resulting system is where Ω ∈ R n is a bounded domain with smooth boundary zΩ.
In system (5a)-(5c), when c � δ � μ, the system without diffusion is so-called the Michaelis-Menten ratio-dependent predator-prey system, which has been studied by many researchers. Kuang and Beretta [37] systematically studied the global behaviors of solutions and obtained some new and significant results, but many important open questions remain to be unsolved. For these open questions, Hsu et al. [38] resolved the global stability of all equilibria in various cases and the uniqueness of limit cycles by transforming the Michaelis-Menten-type ratio-dependent model. Xiao and Ruan [39] investigated the qualitative behavior of the Michaelis-Menten-type ratio-dependent model at the origin in the interior of the first quadrant and confirmed that the origin is indeed a critical point inducing rich and complicated dynamics. Additionally, when the diffusion process is considered, the Michaelis-Menten ratio-dependent predator-prey system with diffusion can produce rich spatial patterns, which makes it a widely studied system for pattern formation [10,[40][41][42][43].
While c < δ, Kovács et al. [44] incorporated delays into system (5a)-(5c) and studied the qualitative behaviour of the system without diffusion. Yun et al. [45] presented an efficient and accurate numerical method for solving system (5a)-(5c) with a Turing instability and studied the existence of nonconstant stationary solutions. Aly et al. [46] studied Turing instability for system (5a)-(5c) and showed that diffusion-driven instability occurs at a certain critical value analytically. In these works, parameters in system (5a)-(5c) are always considered as constants.
However, it seems that there is no research for considering spatial heterogeneity and time-periodic environment in system (5a)-(5c). It is well known that spatial heterogeneity occurs at all scales of the environment [47]. Additionally, interactive populations often live in a fluctuating environment [48], where some environmental conditions such as temperature, light, availability of food, and other resources usually vary in time. Specially, some data depending on season in systems may be periodic functions of time. us, more realistic models to describe ecosystem should be nonautonomous systems with spatial heterogeneity. With this mind, we propose the following system to study effects of spatial heterogeneity and time-periodic environment on prey-predator dynamics: where u(t, x) and v(t, x) represent the densities of the prey and predator, respectively, at a space point x and time t; for simplification, u(t, x) and v(t, x) are rewritten as u and v in the rest of this paper, respectively; r(t, x) is the intrinsic growth rate of prey population; K(t, x) denotes the environmental carrying capacity of prey population; a(t, x) is the capturing rate of the predator; b(t, x) is the half saturation; and e(t, x) denotes the conversion rate. e term c(t, x) describes the specific mortality of predators in absence of prey population, where c(t, x) is the mortality at low density and δ(t, x) is the limiting, maximal mortality. e terms μ 1 Δu(t, x) and μ 2 Δv(t, x) with positive diffusion coefficients μ 1 and μ 2 represent the nonhomogeneous dispersion of the prey and the predator, respectively. Neumann boundary conditions (see equation (6c)) are employed, which characterize the absence of migration.
Here, we assume that prey and predator populations are confined to a fixed bounded space domain Ω ∈ R n with smooth boundary zΩ and Ω � Ω ∪ zΩ. e rest of the paper is organized as follows. In Section 2, some conditions and definitions are given. In Section 3, dynamics of system (6a)-(6c) are studied, including boundedness, permanence, extinction, and periodic solution. Moreover, a series of numerical simulations are carried out for further study of the dynamics of system (6a)-(6c) in Section 4. Finally, the paper ends with conclusion in Section 5.

Preliminaries
Let R, Z, and N be the sets of all real numbers, integers, and positive integers, repectively, and R + � [0, +∞). We assume that the following condition holds throughout the paper: x) are bounded positive-valued functions on R × Ω, continuously differentiable in t and x, and are periodic in t with period τ > 0.
Moreover, for a continuous function ϕ(t, x), we denote for all x ∈ Ω and t ≥ T.
is permanent if there exist positive constants ζ and η such that for every solution with nonnegative initial functions u 0 (x)≢0 and v 0 (x)≢0, there exists a moment of time for all x ∈ Ω and t ≥ t.
Consider the following equations: en, we have the following definition.
To analyze dynamics of system (6a)-(6c), the following results will be needed.

Boundedness.
From the biological and ecological viewpoint, we are always interested in the nonnegative solutions. us, the following theorem is given first in system (6a)-(6c).
Theorem 3. Suppose that the condition (H) holds, then nonnegative and positive quadrants of R 2 are positively invariant for system (6a)-(6c).
Additionally, u is a solution of the following system: From system (6a), we can obtain which implies u(t, x) is a lower solution of system (6a). According to eorem 2, it is obvious that u(t, x) ≥ 0 for all x ∈ Ω and t > 0. Furthermore, due to u 0 (x) ≥ 0(≢0), u(t, x) > 0 holds for all x ∈ Ω and t > 0. us, u(t, x) > 0 holds because u(t, x) is bounded from below by positive function u(t, x).
For system (6b), it can be simply verified that By the similar argument to u(t, x), we can prove the positiveness of v(t, x).
is completes the proof. Based on eorem 3, we will discuss ultimate boundedness of solutions in system (6a)-(6c), and then the following theorem can be obtained. Proof. From system (6a), it can be found that the following inequality holds: Let u(t, x, u 0 ) be a solution of then According to eorem 1, we can get By the uniqueness theorem, it is obvious that the solution u(t, M u ) with initial conditions independent of x does not depend on x for t > 0. erefore, u(t, M u ) is the solution of the following ordinary differential equation: Hence, we have us, there exists a positive constant M 1 in system (6a)-(6c) such that u(t, x) ≤ M 1 , starting with some moment of time.
For predator population v, by system (6b), we have is a solution of the following initial value problem: and erefore, v(t, x, u 0 , v 0 ) is also ultimately bounded. is completes the proof. □ 4 Complexity

Theorem 5. Under the condition (H), if the following inequalities
hold, then system (6a)-(6c) is permanent, i.e., there exist positive constants m i and M i (i � 1, 2) such that any solution of system (6a)-(6c) with nonnegative initial functions for all x ∈ Ω and t > 0. us, for some small ε > 0, we can get initial conditions (u(ε, x, u 0 , v 0 ), v(ε, x, u 0 , v 0 )) separated from zero by the solution on the interval t ≥ ε. Without loss of generality, we assume that min x∈ Ω u 0 (x) � m u , min x∈ Ω v 0 (x) � m v . en, the following inequality holds: Obviously, we can get Consequently, for t ≥ 0, we have us, the solution u(t, x, u 0 , v 0 ) is bounded from below by a solution of the following logistic equation: us, by eorem 1 and condition (24a) and (24b), we have erefore, there exists a positive constant m 1 such that u(t, x, u 0 , v 0 ) ≥ m 1 for t large enough.
By system (6b), the following inequality holds: By a similar analysis to u, is a solution of the following system: According to condition (24b), we can obtain that there exists a positive m 2 such that v(t, x, u 0 , v 0 ) ≥ m 2 for t large enough. us, system (6a)-(6c) is permanent, starting with a certain time.
is completes the proof.

Extinction.
In this section, we will discuss the extinction of predator species, and then the following theorem arrives in system (6a)-(6c).

Theorem 6. If the condition (H) holds, and
Proof. Suppose M v is a fixed positive constant guaranteeing M v ≤ v 0 (x), and v(t, M v ) is the solution of the following initial value problem: By system (6b), we have us, according to eorem 1, we can deduce that is completes the proof.

Periodic Solution.
In this section, we will study the periodic solutions in system (6a)-(6c) by constructing a proper Lyapunov function.

Theorem 7. Under the condition (H), assume that system (6a)-(6c) is permanent, that is, there exist positive constants
N and M such that an arbitrary solution of system (6a)-(6c)

Complexity 5
with nonnegative initial functions not identically equal to zero satisfies the condition: starting with a certain moment of time. If where λ M is the maximal eigenvalue of the following matrix: where en system (6a)-(6c) has a unique and strictly positive τ-periodic solution, which is globally asymptotically stable.
Proof. Let (u(t, x), v(t, x)) and (u(t, x), v(t, x)) be two solutions of system (6a)-(6c) bounded by constants N and M from below and above, respectively. Consider the following function:

Complexity
By condition (36), we have Consider the sequence (u(kτ, en, W(kτ, W 0 ), k ∈ N is compact in the space C(Ω) × C(Ω). Let W be a limit of this sequence, then W(τ, W) � W.
is completes the proof.

Numerical Results
In the previous section, we have obtained some interesting results of system (6a)-(6c). However, due to the complexity of system (6a)-(6c), it becomes much more difficult to provide in-depth analysis. us, here, we perform some numerical simulations to investigate prey-predator dynamics further.
According to eorem 5, when r L − a M /b L > 0 and e L − δ M > 0 holds, system (6a)-(6c) is permanent under condition (H). Figure 1 shows that system (6a)-(6c) is permanent, where r L − a M /b L ≈ 0.1 > 0 and e L − δ M � 0.528 > 0. When e � 0.005 − 0.002 sin(π * t/10), other parameters are the same as the ones in Figure 1, and we can get a numerical solution of system (6a)-(6c) (see Figure 2). It is obvious that predator population v is extinct ultimately, which is consistent with eorem 6 because e M − c L � − 0.008 < 0.
In section 3.4, the existence of periodic solution was discussed, and its stability and uniqueness were analyzed as well. In fact, Figure 1 has shown the existence of a periodic solution. Yet, we here take another set of function corresponding to the parameters of system (6a)-(6c), which is only the periodic function of time t with period 200. e corresponding numerical solutions are shown in Figure 3. Clearly, the numerical solution is periodic in t with the period of 200 (see Figures 3(c) and 3(d)), but it is homogeneous in space (see Figures 3(a) and 3(b)). Compared to Figure 3, we consider another situation that the parameters of system (6a)-(6c) are functions with respect to both time t and space x. We find the solution is still periodic, but it is heterogeneous in space (see Figure 4). It is evident that the spatial heterogeneity is the reason giving rise to the oscillation of the solution in space.

Conclusion
In this paper, we first propose a reaction-diffusion system (6a)-(6c) to describe the interaction between the prey and the predator, where the spatial heterogeneity and the timeperiodic environment are considered. In order to study the boundedness of solution, the positive invariance of system (6a)-(6c) is discussed, and the results demonstrate that nonnegative and positive quadrants of R 2 are always positively invariant for system (6a)-(6c) when the condition (H) holds. Based on this, we find that all solutions of system (6a)-(6c) are ultimately bounded as long as the initial conditions are nonnegative. Also, we discuss the permanence of system (6a)-(6c) and obtain the sufficient conditions. Moreover, we derive the sufficient conditions for the extinction of predator population. Obviously, these conditions are very significant for studies of permanence and extinction of the system. When system (6a)-(6c) is permanent, we discuss the existence of a periodic solution, which suggests that a unique and strictly positive periodic solution with fixed period exists under certain conditions. According to theoretical analysis, some numerical results are given, which show further dynamics in system (6a)-(6c). Results from literature [45,46] indicate that Turing patterns can exist in system (5a)-(5c) (i.e., system (6a)-(6c) without spatial heterogeneity and time-periodic environment). After taking time-periodic environment into account, we find that both prey population and predator population are extinct. However, when the combination of spatial heterogeneity and time-periodic environment is considered, it is demonstrated that both prey population and predator population are permanent, which means that spatial heterogeneity tends to enhance the persistence of prey and predator population. Additionally, when prey population and predator population are permanent, our results show that solutions of system (6a)-(6c) seem to be periodic because of the time-periodic environment and the spatial heterogeneity.
us, we want to emphasize that spatial heterogeneity and time-periodic environment indeed play a significant role in prey-predator dynamics.

Data Availability
e data used to support the findings of this study are included within the article.