Dynamical analysis of a biological resource management model with impulsive releasing and harvesting
© Jiao et al; licensee Springer. 2012
Received: 27 August 2011
Accepted: 11 February 2012
Published: 11 February 2012
In this study, we consider a biological resource management predator-prey model with impulsive releasing and harvesting at different moments. First, we prove that all solutions of the investigated system are uniformly ultimately bounded. Second, the conditions of the globally asymptotic stability predator-extinction boundary periodic solution are obtained. Third, the permanence condition of the investigated system is also obtained. Finally, the numerical simulation verifies our results. These results provide reliable tactic basis for the biological resource management in practice.
Keywordspredator-prey model impulsive releasing impulsive harvesting extinction permanence
Biological resources are renewable resources. Economic and biological aspects of renewable resources management have been considered by Clark . In recent years, the optimal management of renewable resources, which has direct relationship to sustainable development, has been studied extensively by many authors [2–4]. Especially, the predator-prey models with harvesting (or dispersal and competition) are investigated by many articles [5–8]. In general, the exploitation of population should be determined by the economic and biological value of the population. It is the purpose of this article to analyze the exploitation of the predator-prey model with impulsive releasing and harvesting at different moments.
The biological meanings of the parameters in (1.1) can be seen in . Jiao and Chen  consider the mature predator population is harvested continuously. In fact, the population with economic value are harvested discontinuously. It will be arisen at fixed moments or state-dependent moments, that is to say, the releasing population and harvesting population should be occurred at differential moments in . In this article, in order to model the fact of the biological resource management, we investigate a differential equation with two impulses for the biological resource management.
2 The model
where x1(t) and x2(t) are densities of the prey population and the predator population, respectively, r > 0 is the intrinsic growth rate of prey, a > 0 is the coefficient of intraspecific competition, b > 0 is the per-capita rate of predation of the predator, d > 0 is the death rate of predator, c > 0 denotes the product of the per-capita rate of predation and the rate of conversing prey into predator. If rc < ad is satisfied, the predator x2(t) will go extinct and the prey will tend to r/a, that is to say, system (2.1) has boundary equilibrium r/a, 0). If rc > ad is satisfied, system (2.1) has globally asymptotically stable unique positive equilibrium (d/c, rc - ad/cb).
where x(t) denotes the density of the predator population at time t. y(t) denotes the density of the prey population Y at time t. a > 0 denotes the intrinsic growth rate of the prey population X. b > 0 denotes the coefficient of the intraspecific competition in prey population X. β > 0 denotes the per-capita rate predation of the predator population Y. k > 0 denotes product of the per-capita rate and the rate of conversing prey population X into predator population Y. d > 0 denotes the death rate of the predator population Y. 0 < μ1 < 1 denotes the harvesting rate of prey population X at t = (n + l)τ, n ∈ Z+. 0 < μ2 < 1 denotes the harvesting rate of predator population Y at t = (n + l)τ, n ∈ Z+. μ > 0 denotes the released amount of prey population X at t = (n + 1)τ, n ∈ Z+. Δx(t) = x(t+) - x(t), where x(t+) represents the density of prey population X immediately after the impulsive releasing (or harvesting) at time t, while x(t) represents the density of prey population X before the impulsive releasing (or harvesting) at time t. Δy(t) = y(t+) - y(t), where y(t+) represents the density of predator population Y immediately after the impulsive harvesting at time t, while y(t) represents the density of predator population Y before the impulsive harvesting at time t. 0 < l < 1, and τ denotes the period of impulsive effect.
3 The lemmas
Before discussing main results, we will give some definitions, notations and lemmas. Let R+ = [0, ∞), . Denote f = (f1, f2) the map defined by the right hand of system (2.2). Let , then V is said to belong to class V0, if
(i) V is continuous in and , for each , and exists.
(ii) V is locally Lipschitzian in z.
The solution of system (2.2), denote by z(t) = (x(t), y(t)) T , is a piecewise continuous function , z(t) is continuous on and . Obviously, the global existence and uniqueness of solutions of (2.2) is guaranteed by the smoothness properties of f, which denotes the mapping defined by right-side of system (2.2) (see Lakshmikantham ).
Before we have the main results. We need give some lemmas which will be used in the next. Since (dx(t)/dt = 0) whenever x(t) = 0, dy(t)/dt = 0 whenever y(t) = 0, t = nτ, x(nτ+) = (1 - μ1)x(nτ), y(nτ+) = (1 - μ2)y(nτ), and t = (n + l)τ, x((n + l)τ+) = x((n + l)τ) + μ, μ ≥ 0. We can easily have
Lemma 3.2. Suppose z(t) is a solution of system (2.2) with z(0+) ≥ 0, then z(t) ≥ 0 for t ≥ 0. and further z(t) > 0 t ≥ 0 for z(0+) > 0.
Now, we show that all solutions of (2.3) are uniformly ultimately bounded.
Lemma 3.3. There exists a constant M > 0 such that x(t) ≤ M, y(t) ≤ M for each solution (x(t), y(t)) of (2.2) with all t large enough.
So V(t) is uniformly ultimately bounded. Hence, by the definition of V(t), there exists a constant M > 0 such that x(t) ≤ M, y(t) ≤ M for t large enough. The proof is complete.
which can be easily proved to be globally asymptotically stable.
Then, we can derive the following lemma:
4 The dynamics
In this article, we will prove that the predator-extinction periodic solution is globally asymptotically stable and system (2.2) is permanent.
4.1 The extinction
From above discussion, we know that (2.2) has a predator-extinction periodic solution . Then we have following theorem.
hold, then predator-extinction periodic solution of (2.2) is globally asymptotically stable. Where x* is defined as (3.5):
According to the Floquet theory , if | λ2 |< 1, i.e. (4.1) holds, then is locally stable.
So y((n + l + 1)τ+) ≤ y((n + l)τ+)(1 - μ2) exp , hence y((n + l)τ+) ≤ y(lτ+) ρ n and y((n + l)τ+) → 0 as n → ∞. Since 0 < y(t) ≤ y((n + l)τ+)(1 - μ1) for (n + l)τ < t ≤ (n + l + 1)τ, therefore y(t) → 0 as t → ∞.
and A1 = (1-μ1)(a - βε)e(a-βε)τ> 0 and B1 = be(a - βε)lτ[1 + (1 - μ1)(e(a - βε)(1-l)τ- 1)] > 0.
for t large enough, which implies as t → ∞. This completes the proof.
4.2 The permanence
The following study is to investigate the permanence of system (2.2). Before starting this study, we should give the following definition.
Definition 4.2. System (2.2) is said to be permanent if there are constants m, M > 0 (independent of initial value) and a finite time T0 such that for all solutions (x(t), y(t)) with all initial values x(0+) > 0, y(0+) > 0, m ≤ x(t) ≤ M, m ≤ y(t) ≤ M holds for all t ≥ T0. Here T0 may depends on the initial values (x(0+), (y(0+)).
holds, then, system (2.2) is permanent. Where x* is defined as (3.5).
Proof. Let (x(t), y(t)) be a solution of (2.2) with x(0) > 0, y(0) > 0. By Lemma 3.3, we have proved there exists a constant M > 0 (βM < a) such that x(t) ≤ M, y(t) ≤ M for t large enough. We may assume x(t) ≤ M, y(t) ≤ M for t ≥ 0.
- (1)By the condition of Theorem 2, we can select m3 > 0, ε1 > 0 small enough such that , and
where z* is defined as (4.20). We will prove that y(t) < m3 cannot hold for t ≥ 0.
and and .
If y(t) ≥ m3 for t ≥ t1, then our aim is obtained. Hence, we only need to consider those solutions which leave region and reenter it again. Let , there are two possible cases for t*.
which is a contradiction. Let , thus for , we have for . So we have y(t) ≥ m1. The same arguments can be continued since . Hence for all .
Case 2. t ≠ (n + l - 1)τ, n ∈ Z+, then y(t) ≥ m3 for t ∈ [t1, t*), and y(t*) = m3. Suppose , then there are two possible cases for .
Case 2(a). y(t) ≤ m3 for all . Similar to case 1., we can prove that there must be a such that . Here we omit it.
Let , so y(t) ≥ m1 for . For , the same arguments can be continued since y(t) ≥ m1.
Since for , the same arguments can be continued. Hence y(t) ≥ m3 for t ≥ t1. This completes the proof.
From Theorems 4.1 and 4.3, we can easily guess that there must exist an impulsive harvesting predator population threshold . If , the predator-extinction periodic solution of system (2.2) is globally asymptotically stable. If , system (2.2) is permanent. The same discussion can be applied to parameters μ1 and τ. These results show that the impulsive effect plays an important role for the permanence of system (2.2). Our results provide reliable tactic basis for the practically biological resource management.
The authors were grateful to the associate editor, Professor Leonid Berezansky, and the referees for their helpful suggestions that are beneficial to our original article. This study was supported by the Development Project of Nature Science Research of Guizhou Province Department (No. 2010027), the National Natural Science Foundation of China (10961008), and the Science Technology Foundation of Guizhou(2010J2130).
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