ABSTRACT
A stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.
1. Introduction
The construction and study of models for the population dynamics of predator–prey interactions have remained an important area in theoretical ecology since the famous Lotka–Volterra equations. A most crucial element in these models is the functional response, the function that describes the number of prey consumed per predator per unit time for given quantities of prey and predator. In classical models of Lotka–Volterra type, it is assumed that all individuals of a single species have largely similar capabilities to hunt or reproduce. This assumption seems not be realistic for most, if not all, animals and insects. In the real world, there are many species whose individuals have a life history that takes them through two stages, immature and mature, where immatures are raised by their parents, and the rate they attack at prey and the reproductive rate can be ignored (see, for example, [Citation8, Citation10, Citation18, Citation21, Citation24]).
Beginning with the pioneering work of Aiello and Freedman [Citation1] in 1990s, who proposed and studied a single species model with stage structure and discrete delay, stage-structured models with discrete delay have attracted much attention in recent decades, (see, for example, [Citation2, Citation4, Citation6, Citation8–10, Citation16, Citation17]). An important assumption behind those works is that all individuals belonging to the same species take the identical amount of time to become mature, which seems biologically unreasonable since individuals in a population do not necessarily always mature at the same age.
In view of such stage-structured predator–prey model, Gourley and Kuang [Citation12] formulated a general and robust predator–prey model with stage structure with constant maturation time delay and performed a systematic mathematical and computational study. They have shown that there is a window in maturation time-delay parameter that generates sustainable oscillatory dynamics. Thus, they study the following model (1) (1) where and represent prey and mature predator densities, respectively. denotes the immature or juvenile predator density. Here, r is the specific growth rate of the prey and k is its carrying capacity. The parameters b and d are the adult predators' birth and death rates, respectively. In addition, the juveniles suffer a mortality rate of γ (the through-stage death rate) and take τ units of time to mature. The function is the adult predators' functional response and it is assumed to be differentiable and satisfy
An alternative of stage-structured models with discrete delays is to use a distributed delay term allowing for a distribution of maturation times, weighted by a probability density function. Therefore, the stage-structured models of distributed delay type is a generalization of those corresponding model of discrete delay type. There are many different stage-structured models with distributed delay in the literature (see, for example, [Citation3, Citation5, Citation7, Citation12, Citation18, Citation19]).
Al-Omari and Gourley [Citation7] proposed a general model for a single species with stage structure in which the maturation delay is modelled as a distribution. The model takes the form (2) (2) where and , because f is a probability density function. In this system, and denote the numbers of immature (juvenile) and mature (adult) members of the population, respectively, and and are called the birth and death functions and satisfy certain assumptions. The parameter γ measures the juvenile death rate. They considered various aspects of the above system including positivity and also found that the dynamics of the model depends largely on the qualitative form of the birth function, which depends on the total number of adults. They also derived a reaction–diffusion extension of the model, where they found that it again depends largely on the qualitative form of the birth function.
Motivated by the above works, the objective of this paper is to adapt and study system (Equation1(1) (1) ) by introducing the distributed maturation time delay into predator populations and linear harvesting of the immature and mature predator populations and also of prey population. Also, we use general birth and death functions for the mature predator populations. Therefore, we consider the following model: (3) (3) Here, the terms , and are the harvesting efforts of the prey, mature and immature populations, respectively. Note that the term in system (Equation2(2) (2) ) is replaced by the term in system (Equation3(3) (3) ), because is the probability of an individual born at time still being alive and is not harvested at time t. This follows from the assumption that the death and harvesting of immature populations are following a linear law given by the terms and .
Notice that, by the third equation of system (Equation3(3) (3) ), we have (4) (4) that is, is completely determined by and . Therefore, the dynamics of model (Equation3(3) (3) ) are determined by the first two equations. Now, both r and k of system (Equation3(3) (3) ) can be easily scaled off by appropriate rescaling of time and the x variable. Accordingly, system (Equation3(3) (3) ) becomes (5) (5) For initial data of system (Equation5(5) (5) ), we assume that (6) (6) It is straightforward to show that the solutions of system (Equation5(5) (5) ), subject to (Equation6(6) (6) ), satisfy on . This fact is important for both the modelling and the analysis.
In the next sections, we shall prove theorems for the case when the kernels have compact support, that is, for all , for some , and normalized . This implies that no individual ever takes longer than τ units of time to mature. In this case system (Equation5(5) (5) ) becomes, where , (7) (7) Also, in the rest of this paper, we need the following assumptions:
and is strictly increasing for all .
, is strictly increasing bounded for all .
Now, we will prove a theorem giving that the solutions of Equation (Equation7(7) (7) ) are eventually bounded for all time.
Theorem 1.
If holds, then every solution of system (Equation7(7) (7) ) is uniformly bounded.
Proof.
We first prove that is eventually bounded. By positivity of solutions By comparison, where satisfies But . Therefore and so is uniformly bounded.
Let By positivity of solutions, all terms in V are positive. Now so that but since is uniformly bounded, there exists a positive constant M such that Hence Thus, V is uniformly bounded. Since the individual terms in the expression for V are all positive, we deduce that y is uniformly bounded.
We believe that a boundedness result probably holds even if . Because if y got very large the effect on x would be inhibitive, and then low values of x would, in turn, reduce y.
2. Equilibria and their feasibility
The equilibria of our model are determined by setting in system (Equation7(7) (7) ) and solving the resulting algebraic equations.
If , then , so that or . This gives two equilibria the origin and . But it is clear that if , then is the only nonnegative equilibrium of system (Equation7(7) (7) ). If , then a further equilibrium may be possible. The component of such an equilibrium would need to satisfy (8) (8) and we also need , because otherwise would be negative. From the assumptions , one easily sees that the condition for an ecologically relevant equilibrium of the form (distinct from ) is (9) (9) The component of the equilibrium is found from (10) (10)
Note that condition (Equation9(9) (9) ) makes clear how the interior equilibrium , if feasible, depends on the immatures' mortality rate, γ, or the harvesting effort of the immature, , or mature predator populations, , and it depends on the maturation delay as represented by the probability density function and the other parameters. It is automatically satisfied if the parameters γ or or is small enough.
3. Global stability of equilibria
In this section, we shall prove theorems on the global stability of the equilibria , and .
3.1. Global stability of
We shall show that when the equilibria and do not exist, then is globally asymptotically stable.
Theorem 2.
Assume the initial data (Equation6(6) (6) ) and hold. Then as .
Proof.
Consider the functional Note that and if and only if and . Then for t sufficiently large, A direct application of the well-known Lyapunov–LaSalle theorem [Citation17] (Kuang [Citation16, Theorem 2.5.3]) shows that and . The proof of the theorem is complete.
Therefore, our results suggest that the trivial equilibrium is globally stable when the other equilibria are not feasible. That is, we obtained that, if the harvesting effort of the prey population, , is larger than one, then the prey and predator population will go to extinction.
3.2. Global stability of
In this section, we prove a theorem on the global stability of the equilibrium when infant mortality of the predator or harvesting effort of the infant or adult predator is very high, i.e. when . That is, the boundary equilibrium is globally stable if the coexistence equilibrium does not exist.
Theorem 3.
Assume the initial data (Equation6(6) (6) ), and hold. Then as .
Proof.
Since , there exists such that . Also, by positivity of solutions, . This implies that and therefore, there exists such that for . Then, for By comparison, is bounded above by the solution of with, for , . Then since , it follows from [Citation3, Theorem 3] that . Hence .
Next, we shall show that as . Let , then since there exists such that when . Then, for which means that where satisfied From H2, for sufficiently small ϵ, as , where is the positive root of the right-hand side of the last differential equation. Accordingly, Letting , we conclude that We already know that . Consequently, . This completes the proof of the theorem.
3.3. Global stability of
Finally, to prove the stability of we need the following lemma based on Hale and Waltman's famous persistence theory [Citation14]. Consider a complete metric space with metric d. is a continuous semiflow on , that is, a continuous mapping with the following properties: Here, denotes the mapping from to given by . The distance of a point from a subset Y of is defined by Recall that the positive orbit through x is defined as , and its ω-limit set is . Define the stable set of a compact invariant set A as and the particular invariant sets of interest are .
(H3): Assume that is the closure of open set and is nonempty and is the boundary of . Moreover, the -semigroup on satisfies (11) (11)
Lemma 1
[Citation14]
Assume that satisfies (H3) and the following conditions
there exists a such that is compact for
is point dissipative in
is isolated and has acyclic covering
.
Then, is a uniform repeller with respect to that is, there exists an such that for any .
Theorem 4.
Assume that and hold, then system (Equation7(7) (7) ) is uniformly persistent.
Proof.
We begin by showing that the boundary planes of repel the positive solutions to system (Equation7(7) (7) ) uniformly. Let denote the space of continuous functions mapping into . We choose Denote , and , then . Clearly, .
We verify below that the conditions of Lemma 1 are satisfied. We have from the positivity of the solutions that and are invariant, and thus holds. Moreover, by Theorem 1, assumptions and , , conditions (i) and (ii) of Lemma 1 are satisfied. Note that system (Equation7(7) (7) ) has two constant solutions in , , . It is clear that if is a solution of system (Equation7(7) (7) ), then we have , then we get for all , using the second equation of (Equation7(7) (7) ), we get , hence all points in approaches , i.e. . Similarly, we have all points in approach , that is, . Hence, and clearly it is isolated. Noting that , this means that is acyclic, satisfying condition (iii) of Lemma 1.
Now, we show that , . Since Theorem 1 indicates that , we only need to prove . Assume on the contrary, that is, , then there exists a positive solution of system (Equation7(7) (7) ) with . Note that, since for sufficiently small ϵ there exists such that and also, since , then (12) (12) By the second equation of Equation (Equation7(7) (7) ), we have Consider the equation (13) (13) It is easy to obtain that , for Equation (Equation13(13) (13) ). Now, let denotes a positive equilibrium of Equation (Equation13(13) (13) ). Denote , thus Equation (Equation13(13) (13) ) takes the form Therefore, by using Equation (Equation12(12) (12) ) and [Citation3, Theorem 4] we will have , then for all solution of Equation (Equation13(13) (13) ). Thus, this leads to . Contradiction the boundedness of . Then, holds. Thus, all conditions of Lemma 1 are satisfied. Therefore, there exists an such that (14) (14) Thus, system (Equation7(7) (7) ) is uniformly persistent. The proof is complete. Now it follows from [Citation14, Theorem 3.2] that the following theorem holds.
Theorem 5.
Assume that the initial data (Equation6(6) (6) ), and hold, then the equilibrium is globally asymptotically stable.
Thus, the interior equilibrium is globally stable, that is the predator coexists with the prey permanently, if and only if , which is automatically satisfied if the immature species has small mortality rates γ, and/or little harvesting effort among the immature and mature species , y as measured by and , respectively, leading to the permanence of the predator population. Also, we can emphasize that the dynamics of our model depends heavily on the qualitative form of the birth and death functions, which depend on the total number of adults, as well as on the maturation time delay of the predator population as represented by the probability density function .
Acknowledgments
The author is very grateful to the anonymous referee for a careful reading, helpful suggestions and valuable comments which led to the improvement of the manuscript.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
This research is self funded.
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