Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 19, 1-14; http://www.math.u-szeged.hu/ejqtde/

Dynamic analysis of an impulsively controlled predator-prey system

Hunki Baek

Abstract

In this paper, we study an impulsively controlled predator-prey model with Monod-Haldane functional response. By using the Floquet theory, we prove that there exists a stable prey-free solution when the impulsive period is less than some critical value, and give the condition for the permanence of the system. In addition, we show the existence and stability of a positive periodic solution by using bifurcation theory.

Key words: Predator-prey model, Monod-Haldane type functional response, impulsive differential equation, Floquet theory.

2000 AMS Subject Classification. 34A37, 34D23, 34H05, 92D25

1 Introduction

In population dynamics, one of central goals is to understand the dynamical relationship between predator and prey. One important component of the predator-prey relationship is the predator’s rate of feeding on prey, i.e., the so-called predator’s functional response. Functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. Holling [7] gave three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. These functional responses are monotonic in the first quadrant. But, some experiments and observations indicate that a non-monotonic response occurs at a level: when the nutrient concentration reaches a high level an inhibitory effect on the specific growth rate may occur. To model such an inhibitory effect, the authors in [1, 14] suggested a function

p(x) = \frac{αx}{b + x^{2}}

Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea, (hkbaek@knu.ac.kr).

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (No.2009-0073916).

called the Monod-Haldane function, or Holling type-IV function. We can write a predatorprey model with Monod-Haldane type functional response as follows.

\{ \begin{matrix}x′(t) = ax(t)(1 − \frac{x(t)}{K}) − \frac{cx(t)y(t)}{b + x^{2}(t)}, \\ y′(t) = −Dy(t) + \frac{mx(t)y(t)}{b + x^{2}(t)}, \end{matrix}  (1)

where x(t) and y(t) represent population densities of prey and predator at time t. All parameters are positive constants. Usually, a is the intrinsic growth rate of the prey, K is the carrying capacity of the prey, the constant D is the death rate of the predator, m is the rate of conversion of a consumed prey to a predator and b measures the level of prey interference with predation.

As Cushing [5] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits or harvesting seasons and so on). Such perturbations were often treated continually. But, there are still some other perturbations such as fire, flood, etc, that are not suitable to be considered continually. These impulsive perturbations bring sudden changes to the system.

In this paper, with the idea of impulsive perturbations, we consider the following predator-prey model with periodic constant impulsive immigration of the predator and periodic harvesting on the prey.

\{ \begin{matrix}\begin{matrix}x′(t) = ax(t)(1 − \frac{x(t)}{K}) − \frac{cx(t)y(t)}{b + x^{2}(t)}, \\ y′(t) = −Dy(t) + \frac{mx(t)y(t)}{b + x^{2}(t)}, \end{matrix} \} t ≠ nT, \\ \begin{matrix}x(t^{+}) = (1 − p_{1})x(t), \\ y(t^{+}) = (1 − p_{2})y(t) + q, \end{matrix} \} t = nT, \\ (x(0^{+}), y(0^{+})) = x_{0}, \end{matrix}  (2)

where T is the period of the impulsive immigration or stock of the predator, p_{i}(0 ≤ p_{i} < 1, = 1, 2) are the harvesting control parameters, q is the size of immigration or stock of the predator. This model is an example of impulsive differential equations whose theories and applications were greatly developed by the efforts of Bainov and Lakshmikantham et al. [4, 8].

Recently, many researchers have intensively investigated systems with impulsive perturbations (cf, [2, 3, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21]). Most of such systems have dealt with impulsive harvesting and immigration of predators at different fixed times. On the contrary, here we consider the impulsive harvesting and immigration at the same time in our model which has not been studied well until now.

The main purpose of this paper is to study the dynamics of the system (2).

The organization of this paper is as follows. In the next section, we introduce some notations and lemmas related to impulsive differential equations which are used in this paper. In Section 3, we show the stability of prey-free periodic solutions and give a sufficient condition for the permanence of system (2) by applying the Floquet theory and the comparison theorem. In Section 4, we show the existence of nontrivial periodic solutions via the bifurcation theorem. Finally, in conclusion, we give a bifurcation diagram that shows the system has various dynamic aspects including chaos.

2 Preliminaries

Let ℝ_{+} = [0, ∞) and ℝ_{+}^{2} = {x = (x, y) ∈ ℝ^{2} : x, y ≥ 0}. Denote ℕ the set of all of nonnegative integers and f = (f_{1}, f_{2})^{T} the right hand of (2). Let V : ℝ_{+} × ℝ_{+}^{2} → ℝ_{+}, then V is said to be in a class V_{0} if

(1)V is continuous on(nT, (n + 1)T] × ℝ_{+}^{2}, and lim_{\begin{matrix}(t,y)→(nT,x) \\ t>nT \end{matrix}} V (t, y) = V (nT^{+}, x) exists.
(2)V is locally Lipschitz in x.

Definition 2.1. Let V ∈ V_{0}, (t, x) ∈ (nT, (n + 1)T] × ℝ_{+}^{2}. The upper right derivatives of V (t, x) with respect to the impulsive differential system (2) is defined as

D^{+}V (t, x) = lim sup_{h→0^{+}} \frac{1}{h}[V (t + h, x + hf(t, x)) − V (t, x)].

Remarks 2.1. (1) The solution of the system (2) is a piecewise continuous function x : ℝ_{+} → ℝ_{+}^{2}, x(t) is continuous on (nT, (n + 1)T], n ∈ ℕ and x(nT^{+}) = lim_{t→nT^{+}} x(t) exists.

(2) The smoothness properties of f guarantee the global existence and uniqueness of solution of the system (2). (See [8] for the details).

We will use the following important comparison theorem on an impulsive differential equation [8].

Lemma 2.2. (Comparison theorem) Suppose V ∈ V_{0} and

\{ \begin{matrix}D^{+}V (t, x) ≤ g(t, V (t, x)), t ≠ nτ \\ V (t, x(t^{+})) ≤ ψ_{n}(V (t, x)), t = nτ, \end{matrix}  (3)

g : ℝ_{+} × ℝ_{+} → ℝ is continuous on (nτ, (n + 1)τ] × ℝ_{+} and for u(t) ∈ ℝ_{+}, n ∈ ℕ, lim_{(t,y)→(nτ^{+},u)} g(t, y) = g(nτ^{+}, u) exists, ψ_{n} : ℝ_{+} → ℝ_{+} is non-decreasing. Let r(t) be the maximal solution of the scalar impulsive differential equation

\{ \begin{matrix}u′(t) = g(t, u(t)), t ≠ nτ, \\ u(t^{+}) = ψ_{n}(u(t)), t = nτ, \\ u(0^{+}) = u_{0}, \end{matrix}  (4)

existing on [0, ∞). Then V (0^{+}, x_{0}) ≤ u_{0} implies that V (t, x(t)) ≤ r(t), t ≥ 0, where x(t) is any solution of (3).

Note that \frac{dx}{dt} = \frac{dy}{dt} = 0 whenever x(t) = y(t) = 0, t ≠ nT. So, we can easily show the following lemma.

Lemma 2.3. Let x(t) = (x(t), y(t)) be a solution of (2).

(1) If x(0^{+}) ≥ 0 then x(t) ≥ 0 for all t ≥ 0.

(2) If x(0^{+}) > 0 then x(t) > 0 for all t ≥ 0.

Now, we show that all solutions of (2) are uniformly ultimately bounded.

Lemma 2.4. There is an M > 0 such that x(t), y(t) ≤ M for all t large enough, where (x(t), y(t)) is a solution of (2).

Proof. Let x(t) = (x(t), y(t)) be a solution of (2) and let V (t, x) = mx(t) + cy(t). Then V ∈ V_{0}, and if t ≠ nT

D^{+}V + βV = −\frac{ma}{K}x^{2}(t) + m(a + β)x(t) + c(β − D)y(t).  (5)

Clearly, the right hand of (5), is bounded when 0 < β < D. When t = nT, V (nT^{+}) = mx(nT^{+}) + cy(nT^{+}) = (1 − p_{1})mx(nT) + (1 − p_{2})cy(nT) + cq ≤ V (nT) + cq. So we can choose 0 < β_{0} < D and M_{0} > 0 such that

\{ \begin{matrix}D^{+}V ≤ −β_{0}V + M_{0}, t ≠ nT, \\ V (nT^{+}) ≤ V (nT) + cq. \end{matrix}  (6)

From Lemma 2.2, we can obtain that

V (t) ≤ (V (0^{+})−\frac{M_{0}}{β_{0}} ) exp(−β_{0}t) + \frac{cq(1 − exp(−(n + 1)β_{0}T))}{1 − exp(−β_{0}T)}exp(−β_{0}(t−nT))+\frac{M_{0}}{β_{0}}

for t ∈ (nT, (n + 1)T]. Therefore, V (t) is bounded by a constant for sufficiently large t. Hence there is an M > 0 such that x(t) ≤ M, y(t) ≤ M for a solution (x(t), y(t)) with all t large enough.

Now, we consider the following impulsive differential equation.

\{ \begin{matrix}y′(t) = −Dy(t), t ≠ nT, \\ y(t^{+}) = (1 − p_{2})y(t) + q, t = nT, \\ y(0^{+}) = y_{0}. \end{matrix}  (7)

We can easily obtain the following results.

Lemma 2.5. (1) y^{∗}(t) = \frac{q exp(−D(t − nT))}{1 − (1 − p_{2}) exp(−DT)}, t ∈ (nT, (n + 1)T], n ∈ ℕ is a

positive periodic solution of (7) with the initial value y^{∗}(0^{+}) = \frac{q}{1 − (1 − p_{2}) exp(−DT)} .

(2) y(t) = (1 − p_{2})^{n+1}(y(0^{+}) − \frac{q exp(−DT)}{1 − (1 − p_{2}) exp(−DT)}) exp(−Dt) + y^{∗}(t) is the

solution of (7) with y_{0} ≥ 0, t ∈ (nT, (n + 1)T] and n ∈ ℕ.

(3) All solutions y(t) of (2) with y_{0} ≥ 0 tend to y^{∗}(t). i.e., |y(t) − y^{∗}(t)| → 0 as t → ∞.

It is from Lemma 2.5 that the general solution y(t) of (7) can be identified with the positive periodic solution y^{∗}(t) of (7) for sufficiently large t and we can obtain the complete expression for the prey-free periodic solution of (2)

(0, y^{∗}(t)) = (0, \frac{q exp(−D(t − nT))}{1 − (1 − p_{2}) exp(−DT)}) for t ∈ (nT, (n + 1)T].

3 Extinction and permanence

Now, we present a condition which guarantees local stability of the prey-free periodic solution (0, y^{∗}(t)).

Theorem 3.1. The prey-free solution (0, y^{∗}(t)) is locally asymptotically stable if

aT + ln(1 − p_{2}) < \frac{cq(1 − exp(−DT))}{bD(1 − (1 − p_{2}) exp(−DT))}.

Proof. The local stability of the periodic solution (0, y^{∗}(t)) of (2) may be determined by considering the behavior of small amplitude perturbations of the solution. Let (x(t), y(t)) be any solution of (2). Define x(t) = u(t), y(t) = y^{∗}(t) + v(t). Then they may be written as

(\begin{matrix}u(t) \\ v(t) \end{matrix}) = Φ(t) (\begin{matrix}u(0) \\ v(0) \end{matrix}) , 0 ≤ t ≤ T,  (8)

where Φ(t) satisfies

\frac{dΦ}{dt} = (\begin{matrix}a − \frac{c}{b}y^{∗}(t) & 0 \\ \frac{m}{b}y^{∗}(t) & −D \end{matrix}) Φ(t)  (9)

and Φ(0) = I, the identity matrix. The linearization of the third and fourth equation of (2) becomes

(\begin{matrix}u(nT^{+}) \\ v(nT^{+}) \end{matrix}) = (\begin{matrix}1 − p_{1} & 0 \\ 0 & 1 − p_{2} \end{matrix}) (\begin{matrix}u(nT) \\ v(nT) \end{matrix}) .  (10)

Note that all eigenvalues of S = (\begin{matrix}1 − p_{1} & 0 \\ 0 & 1 − p_{2} \end{matrix}) Φ(T) are μ_{1} = exp(−DT) < 1 and

μ_{2} = (1 − p_{2}) exp(∫_{0}^{T} a − \frac{c}{b}y^{∗}(t)dt). Since

∫_{0}^{T} y^{∗}(t)dt = \frac{q(1 − exp(−DT))}{D(1 − (1 − p_{2}) exp(−DT))},

we have

μ_{2} = (1 − p_{2}) exp(aT − \frac{cq(1 − exp(−DT))}{bD(1 − (1 − p_{2}) exp(−DT))}).

By Floquet Theory in [4], (0, y^{∗}(t)) is locally asymptotically stable if |μ_{2}| < 1.i.e.,

aT < \frac{cq(1 − exp(−DT))}{bD(1 − (1 − p_{2}) exp(−DT))} + ln \frac{1}{1 − p_{2}} .

Definition 3.1. The system (2) is permanent if there exist M ≥ m > 0 such that, for any solution (x(t), y(t)) of (2) with x_{0} > 0,

m ≤ lim_{t→∞} inf x(t) ≤ lim_{t→∞} sup x(t) ≤ M and m ≤ lim_{t→∞} inf y(t) ≤ lim_{t→∞} sup y(t) ≤ M.

Theorem 3.2. The system (2) is permanent if

aT + ln(1 − p_{2}) > \frac{cq(1 − exp(−DT))}{bD(1 − (1 − p_{2}) exp(−DT))}.

Proof. Let (x(t), y(t)) be any solution of (2)with x_{0} > 0. From Lemma 2.4, we may as-

sume that x(t) ≤ M, y(t) ≤ M , t ≥ 0 and M > \frac{ab}{c}. Let m_{2} = \frac{q exp(−DT)}{1 − (1 − p_{2}) exp(−DT)}−

ϵ_{2}, ϵ_{2} > 0. From Lemma 2.5, clearly we have y(t) ≥ m_{2} for all t large enough. Now we shall find an m_{1} > 0 such that x(t) ≥ m_{1} for all t large enough. We will do this in the following two steps.

(Step 1) Since

aT > \frac{cq(1 − exp(−DT))}{bD(1 − (1 − p_{2}) exp(−DT))} + ln \frac{1}{1 − p_{2}} ,

we can choose m_{3} > 0, ϵ_{1} > 0 small enough such that δ = \frac{mm_{3}}{b + m_{3}} < D and R =

(1 − p_{2}) exp(aT − \frac{a}{K} T m_{3} − \frac{cq(1 − exp(−DT))}{bD(1 − (1 − p_{2}) exp(−DT))} − \frac{cϵ_{1}}{b} T ) > 1. Suppose that

x(t) < m_{3} for all t. Then we get y′(t) ≤ y(t)(−D + δ) from above assumptions.

By Lemma 2.2, we have y(t) ≤ u(t) and u(t) → u^{∗}(t), t → ∞, where u(t) is the solution of

\{ \begin{matrix}u′(t) = (−D + δ)u(t), t ≠ nT, \\ u(t^{+}) = (1 − p_{2})u(t) + q, t = nT, \\ u(0^{+}) = y_{0}, \end{matrix}  (11)

and u^{∗}(t) = \frac{q exp((−D + δ)(t − nT))}{1 − (1 − p_{2}) exp((−D + δ)T)}, t ∈ (nT, (n+1)T]. Then there exists T_{1} > 0

such that y(t) ≤ u(t) ≤ u^{∗}(t) + ϵ_{1} and

x′(t) = x(t)(a − \frac{a}{K} x(t) − \frac{cy(t)}{b + x^{2}(t)})

≥ x(t)(a − \frac{a}{K} m_{3} − \frac{c}{b}y(t))

≥ x(t)(a − \frac{a}{K} m_{3} − \frac{c}{b}(u^{∗}(t) + ϵ_{1})) for t ≥ T_{1}.

Let N_{1} ∈ ℕ and N_{1}T ≥ T_{1}. We have, for n ≥ N_{1}

\begin{cases}x′(t) & ≥ x(t)(a − \frac{a}{K} m_{3} − \frac{c}{b}(u^{∗}(t) + ϵ_{1})), t ≠ nT, \\ x(t^{+}) & = (1 − p)x(t), t = nT. \end{cases}  (12)

Integrating (12) on (nT, (n + 1)T](n ≥ N_{1}), we obtain

x((n + 1)T) ≥ x(nT^{+}) exp(∫_{nT}^{(n+1)T} a − \frac{a}{K} m_{3} − \frac{c}{b}(u^{∗}(t) + ϵ_{1})dt) = x(nT)R.

Then x((N_{1} + k)T) ≥ x(N_{1}T)R^{k} → ∞ as k → ∞ which is a contradiction. Hence there exists a t_{1} > 0 such that x(t_{1}) ≥ m_{3}.

(Step 2) If x(t) ≥ m_{3} for all t ≥ t_{1}, then we are done. If not, we may let t^{∗} = inf_{t>t_{1}}{x(t) < m_{3}}.Then x(t) ≥ m_{3} for t ∈ [t_{1}, t^{∗}] and, by the continuity of x(t), we have x(t^{∗}) = m_{3}. In this step, we have only to consider two possible cases.

Case (1) t^{∗} = n_{1}T for some n_{1} ∈ ℕ. Then (1−p_{1})m_{3} ≤ x(t^{∗+}) = (1−p_{1})x(t^{∗}) < m_{3}.

Select n_{2}, n_{3} ∈ ℕ such that (n_{2} − 1)T > \frac{ln(\frac{ϵ_{1}}{M+q})}{−d + δ} and (1 − p_{1})^{n_{2}}R^{n_{3}} exp(n_{2}σT) >

(1−p_{1})^{n_{2}}R^{n_{3}} exp((n_{2}+1)σT) > 1, where σ = a−\frac{a}{K} m_{3}−\frac{c}{b}M < 0. Let T′ = n_{2}T +n_{3}T.

In this case we will show that there exists t_{2} ∈ (t^{∗}, t^{∗} + T′] such that x(t_{2}) ≥ m_{3}. Otherwise, by (11) with u(t^{∗+}) = y(t^{∗+}), we have

u(t) = (1−p_{2})^{n_{1}+1}(u(t^{∗+})−\frac{q exp((−D + δ)T)}{1 − (1 − p_{2}) exp((−D + δ)T)}) exp((−D+δ)(t−t^{∗}))+u^{∗}(t)

for (n−1)T < t ≤ nT and n_{1}+1 ≤ n ≤ n_{1}+1+n_{2}+n_{3}. So we get |u(t)−u^{∗}(t)| ≤ (M + q) exp((−D + δ)(t − t^{∗})) < ϵ_{1} and y(t) ≤ u(t) ≤ u^{∗}(t) + ϵ_{1} for t^{∗} + n_{2}T ≤ t ≤ t^{∗} + T′. Thus we obtain the same equation as (12)

for t ∈ [t^{∗} + n_{2}T, t^{∗} + T′]. As in step 1, we have

x(t^{∗} + T′) ≥ x(t^{∗} + n_{2}T)R^{n_{3}}.

Since y(t) ≤ M, we have

\{ \begin{matrix}x′(t) ≥ x(t)(a − \frac{a}{K} m_{3} − \frac{c}{b}M) = σx(t), t ≠ nT, \\ x(t^{+}) = (1 − p_{1})x(t), t = nT, \end{matrix}  (13)

for t ∈ [t^{∗}, t^{∗} + n_{2}T]. Integrating (13) on [t^{∗}, t^{∗} + n_{2}T] we have

\begin{aligned}x((t^{∗} + n_{2}T)) & ≥ m_{3} exp(σn_{2}T) \\ & ≥ m_{3}(1 − p_{1})^{n_{2}} exp(σn_{2}T) > m_{3}. \end{aligned}

Thus x(t^{∗} + T′) ≥ m_{3}(1 − p_{1})^{n_{2}} exp(σn_{2}T)R^{n_{3}} > m_{3} which is a contradiction. Now, let t̄= inf_{t>t^{∗}}{x(t) ≥ m_{3}}. Then x(t) ≤ m_{3} for t^{∗} ≤ t < t̄and x(t̄) = m_{3}. For t ∈ [t^{∗}, t̄), suppose t ∈ (t^{∗} + (k − 1)T, t^{∗} + kT], k ∈ ℕ and k ≤ n_{2} + n_{3}. So, we have, for t ∈ [t^{∗}, t̄), from (13) we obtain x(t) ≥ x(t^{∗+})(1 − p_{1})^{k−1} exp((k − 1)σT) exp(σ(t − (t^{∗} + (k − 1)T))) ≥ m_{3}(1 − p_{1})^{k} exp(kσT) ≤ m_{3}(1 − p_{1})^{n_{2}+n_{3}} exp(σ(n_{2} + n_{3})T) ≡ m′_{1}.

Case (2) t^{∗} ≠ nT, n ∈ ℕ. Then x(t) ≥ m_{3} for t ∈ [t_{1}, t^{∗}) and x(t^{∗}) = m_{3}. Suppose that t^{∗} ∈ (n′_{1}T, (n′_{1} + 1)T) for some n′_{1} ∈ ℕ. There are two possible cases.

Case(2(a)) x(t) < m_{3} for all t ∈ (t^{∗}, (n′_{1} + 1)T]. In this case we will show that there exists t_{2} ∈ [(n′_{1} + 1)T, (n′_{1} + 1)T + T′] such that x_{2}(t_{2}) ≥ m_{3}. Suppose not, i.e., x(t) < m_{3}, for all t ∈ [(n′_{1} + 1)T, (n′_{1} + 1 + n_{2} + n_{3})T]. Then x(t) < m_{3} for all t ∈ (t^{∗}, (n′_{1} + 1 + n_{2} + n_{3})T]. By (11) with u((n′_{1} + 1)T^{+}) = y((n′_{1} + 1)T^{+}), we have

u(t) = (u((n′_{1}+1)T^{+})−\frac{q exp(−D + δ)}{1 − (1 − p_{2}) exp(−D + δ)}) exp((−D+δ)(t−(n′_{1}+1)T))+u^{∗}(t)

for t ∈ (nT, (n + 1)T], n′_{1} + 1 ≤ n ≤ n′_{1} + n_{2} + n_{3}. By a similar argument as in (step 1), we have

x((n′_{1} + 1 + n_{2} + n_{3})T) ≥ x_{2}((n′_{1} + 1 + n_{2})T)R^{n_{3}}.

It follows from (13) that

x((n′_{1} + 1 + n_{2})T) ≥ m_{3}(1 − p)^{n_{2}+1} exp(σ(n_{2} + 1)T).

Thus x((n′_{1} + 1 + n_{2} + n_{3})T) ≥ m_{3}(1 − p)^{n_{2}+1} exp(σ(n_{2} + 1)T)R^{n_{3}} > m_{3} which is a contradiction. Now, let t̄ = inf_{t>t^{∗}}{x(t) ≥ m_{3}}. Then x(t) ≤ m_{3} for t^{∗} ≤ t < t̄ and x(t̄) = m_{3}. For t ∈ [t^{∗}, t̄), suppose t ∈ (n′_{1}T + (k′ + 1)T, n′_{1}T + k′T], k′ ∈ ℕ, k′ ≤ 1 + n_{2} + n_{2}, we have x(t) ≥ m_{3}(1 − p)^{1+n_{2}+n_{3}} exp(σ(1 + n_{2} + n_{3})T) ≡ m_{1}. Since m_{1} < m′_{1}, so x(t) ≥ m_{1} for t ∈ (t^{∗}, t̄).

Case (2(b)) There is a t′ ∈ (t^{∗}, (n′_{1}+1)T] such that x_{2}(t′) ≥ m_{3}. Let t̂= inf_{t>t^{∗}}{x(t) ≥ m_{3}}. Then x(t) ≤ m_{3} for t ∈ [t^{∗}, t̂) and x(t̂) = m_{3}. Also, (13) holds for t ∈ [t^{∗}, t̂). Integrating the equation on [t^{∗}, t)(t^{∗} ≤ t ≤ t̂), we can get that x(t) ≥ x(t^{∗}) exp(σ(t − t^{∗})) ≥ m_{3} exp(σT) ≥ m_{1}. Thus in both cases a similar argument can be continued since x(t) ≥ m_{1} for some t > t_{1}. This completes the proof.

Remarks 3.3. Define G(T) = aT + ln(1 − p_{2}) − \frac{cq(1 − exp(−DT))}{bD(1 − (1 − p_{2}) exp(−DT))}. Since

G(0) = ln(1 − p_{2}) < 0, lim_{T →∞} G(T) = ∞ and

G′′(T) = \frac{cqp_{2} exp(−DT)(1 + (1 − p_{2}) exp(−DT))}{b^{4}D(1 − (1 − p_{2}) exp(−DT))^{3}} > 0,  (14)

so we have that G(T) = 0 has a unique positive solution T^{∗}. From Theorem 3.1 and Theorem 3.2, we know that the prey-free periodic solution is locally asymptotically stable if T < T^{∗} and otherwise, the prey and predator can coexist. Thus T^{∗} plays the role of a critical value that discriminates between stability and permanence.

4 Existence and stability of a positive periodic solution

Now, we deal with a problem of the bifurcation of the nontrivial periodic solution of the system (2), near (0, y^{∗}(t)). The following theorem establishes the existence of a positive periodic solution of the system (2) near (0, y^{∗}(t)).

Theorem 4.1. The system (2) has a positive periodic solution which is supercritical if T > T^{∗}.

Proof. We will use the results of [6, 9, 11] to prove this theorem. To use theorems of [6, 9, 11], it is convenient for the computation to exchange the variables of x and y and change the period T to τ. Thus the system (2) becomes as follows

\{ \begin{matrix}\begin{matrix}x′(t) = −Dx(t) + \frac{my(t)x(t)}{b + y^{2}(t)}, \\ y′(t) = ay(t)(1 − \frac{y(t)}{K}) − \frac{cy(t)x(t)}{b + y^{2}(t)}, \end{matrix} \} t ≠ nτ, \\ \begin{matrix}x(t^{+}) = (1 − p_{2})x(t) + q, \\ y(t^{+}) = (1 − p_{1})y(t), \end{matrix} \} t = nτ. \end{matrix}  (15)

Let Φ be the flow associated with (15). We have X(t) = Φ(t, x_{0}), 0 < t ≤ τ, where x_{0} = (x(0), y(0)). The flow Φ applies up to time τ. So, X(τ) = Φ(τ, x_{0}). We will use all notations in [9]. Note that

F_{1}(x, y) = −Dx + \frac{myx}{b + y^{2}} , F_{2}(x, y) = ay(1 − \frac{y}{K}) − \frac{cyx}{b + y^{2}} ,
Θ_{1}(x, y) = (1 − p_{2})x + q, Θ_{2}(x, y) = (1 − p_{1})y, ζ(t) = (y^{∗}(t), 0),
\frac{∂Φ_{1}(t, x_{0})}{∂x} = exp(∫_{0}^{t} \frac{∂F_{1}(ζ(r))}{∂x}dr), \frac{∂Φ_{2}(t, x_{0})}{∂y} = exp(∫_{0}^{t} \frac{∂F_{2}(ζ(r))}{∂y}dr),
\frac{∂Φ_{1}(t, x_{0})}{∂y} = ∫_{0}^{t} exp(∫_{ν}^{t} \frac{∂F_{1}(ζ(r))}{∂x}dr)\frac{∂F_{1}(ζ(ν))}{∂y} exp(∫_{0}^{ν} \frac{∂F_{2}(ζ(r))}{∂y}dr)dν.

Then

\frac{∂Θ_{1}}{∂y} = \frac{∂Θ_{2}}{∂x} = 0, \frac{∂Θ_{1}}{∂x} = 1 − p_{2}, \frac{∂Θ_{2}}{∂y} = 1 − p_{1}, \frac{∂^{2}Θ_{i}}{∂x∂y} = 0, i = 1, 2 and \frac{∂^{2}Θ_{2}}{∂y^{2}} = 0.

Now, we can compute

d′_{0} = 1 − (\frac{∂Θ_{2}}{∂y} · \frac{∂Φ_{2}}{∂y} )_{(τ_{0},x_{0})} = 1 − (1 − p_{1}) exp(∫_{0}^{τ_{0}} a − \frac{c}{b}y^{∗}(r)dr),

where τ_{0} is the root of d′_{0} = 0. Actually, we know that τ_{0} = T^{∗}.

a′_{0} = 1 − (\frac{∂Θ_{1}}{∂x} · \frac{∂Φ_{1}}{∂x} )_{(τ_{0},x_{0})} = 1 − (1 − p_{2}) exp(−Dτ_{0}) > 0,

b′_{0} = −(\frac{∂Θ_{1}}{∂x} · \frac{∂Φ_{1}}{∂y} + \frac{∂Θ_{1}}{∂y} · \frac{∂Φ_{2}}{∂y} )_{(τ_{0},x_{0})} = −(\frac{∂Φ_{1}}{∂y} )_{(τ_{0},x_{0})}

= − ∫_{0}^{τ_{0}} exp(∫_{ν}^{τ_{0}} \frac{∂F_{1}(ζ(r))}{∂x}dr)\frac{m}{b}y^{∗}(ν) exp(∫_{0}^{ν} \frac{∂F_{2}(ζ(r))}{∂y}dr)dν < 0,

\frac{∂^{2}Φ_{2}(t, x_{0})}{∂y∂x} = ∫_{0}^{t} exp(∫_{ν}^{t} \frac{∂F_{2}(ζ(r))}{∂y}dr)\frac{∂^{2}F_{2}(ζ(ν))}{∂y∂x} exp(∫_{0}^{ν} \frac{∂F_{2}(ζ(r))}{∂y}dr)dν,
\frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂y∂x} = −\frac{c}{b} ∫_{0}^{τ_{0}} exp(∫_{ν}^{τ_{0}} \frac{∂F_{2}(ζ(r))}{∂y}dr) exp(∫_{0}^{ν} \frac{∂F_{2}(ζ(r))}{∂y}dr)dν < 0,
\frac{∂^{2}Φ_{2}(t, x_{0})}{∂y^{2}} = ∫_{0}^{t} exp(∫_{ν}^{t} \frac{∂F_{2}(ζ(r))}{∂y}dr)\frac{∂^{2}F_{2}(ζ(ν))}{∂y^{2}} exp(∫_{0}^{ν} \frac{∂F_{2}(ζ(r))}{∂y}dr)dν

+ ∫_{0}^{t}[exp(∫_{ν}^{t} \frac{∂F_{2}(ζ(r))}{∂y}dr)\frac{∂^{2}F_{2}(ζ(ν))}{∂y∂x}]

× [∫_{0}^{ν} exp(∫_{θ}^{ν} \frac{∂F_{1}(ζ(r))}{∂x}dr)\frac{∂F_{1}(ζ(θ))}{∂y} exp(∫_{0}^{θ} \frac{∂F_{2}(ζ(r))}{∂y}dr)dθ]dν,

EJQTDE, 2010 No. 19, p. 10

\frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂y^{2}} = − \frac{2a}{K} ∫_{0}^{τ_{0}} exp(∫_{ν}^{τ_{0}} \frac{∂F_{2}(ζ(r))}{∂y}dr) exp(∫_{0}^{ν} \frac{∂F_{2}(ζ(r))}{∂y}dr)dν

− \frac{cm}{b^{2}} ∫_{0}^{τ_{0}}[exp(∫_{ν}^{τ_{0}} \frac{∂F_{2}(ζ(r))}{∂y}dr)]

× [∫_{0}^{ν} exp(∫_{θ}^{ν} \frac{∂F_{1}(ζ(r))}{∂x}dr)y^{∗}(θ) exp(∫_{0}^{θ} \frac{∂F_{2}(ζ(r))}{∂y}dr)dθ]dν < 0,

\frac{∂^{2}Φ_{2}(t, x_{0})}{∂y∂τ} = \frac{∂F_{2}(ζ(t))}{∂y} exp(∫_{0}^{t} \frac{∂F_{2}(x_{p}(r), 0)}{∂y}dr),
\frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂y∂τ} = \frac{∂F_{2}(ζ(τ_{0}))}{∂y} exp(∫_{0}^{τ_{0}} \frac{∂F_{2}(x_{p}(r), 0)}{∂y}dr)

= (a − \frac{c}{b}y^{∗}(τ_{0})) exp(∫_{0}^{τ_{0}}(a − \frac{c}{b}y^{∗}(r))dr),

\frac{∂Φ_{1}(τ_{0}, x_{0})}{∂τ} = ẏ^{∗}(τ_{0}) = −\frac{q exp(−Dτ_{0})}{D(1 − (1 − p_{2}) exp(−Dτ_{0}))} < 0,

C = − 2\frac{∂^{2}Θ_{2}}{∂x∂y} (−\frac{b′_{0}}{a′_{0}} · \frac{∂Φ_{1}(τ_{0}, x_{0})}{∂x} + \frac{∂Φ_{1}(τ_{0}, x_{0})}{∂y})\frac{∂Φ_{2}(τ_{0}, x_{0})}{∂y}

− \frac{∂^{2}Θ_{2}}{∂y^{2}} (\frac{∂Φ_{2}(τ_{0}, x_{0})}{∂y})^{2} + 2\frac{∂Θ_{2}}{∂y} · \frac{b′_{0}}{a′_{0}} · \frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂y∂x} − \frac{∂Θ_{2}}{∂y} · \frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂y^{2}}

= 2(1 − p_{1})\frac{b′_{0}}{a′_{0}} \frac{∂^{2}Φ_{2}}{∂x∂y} − (1 − p_{1})\frac{∂^{2}Φ_{2}}{∂y^{2}} > 0,

B = − \frac{∂^{2}Θ_{2}}{∂x∂y} (\frac{∂Φ_{1}(τ_{0}, x_{0})}{∂τ} + \frac{∂Φ_{1}(τ_{0}, x_{0})}{∂x} · \frac{1}{a′_{0}} · \frac{∂Θ_{1}}{∂x} · \frac{∂Φ_{1}(τ_{0}, X_{0})}{∂τ})\frac{∂Φ_{2}(τ_{0}, x_{0})}{∂y}

− \frac{∂Θ_{2}}{∂y} (\frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂x∂y} · \frac{1}{a′_{0}} · \frac{∂Θ_{1}}{∂x} · \frac{∂Φ_{1}(τ_{0}, x_{0})}{∂τ} + \frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂τ∂y})

= −(1 − p_{1})(\frac{∂^{2}Φ_{2}(τ_{0}, x_{0})}{∂x∂y} · \frac{1}{a′_{0}} · \frac{∂Φ_{1}(τ_{0}, x_{0})}{∂τ}

+ (a − \frac{c}{b}y^{∗}(τ_{0})) exp ∫_{0}^{τ_{0}}(a − \frac{c}{b}y^{∗}(r))dr).

To determine the sign of B, let ϕ(t) = a − \frac{c}{b}y^{∗}(t). Then we obtain that

ϕ′(t) = \frac{cDq exp(−Dt)}{b(1 − (1 − p_{2}) exp(−Dτ_{0}))} > 0

and so ϕ(t) is strictly increasing. Since

∫_{0}^{τ_{0}} ϕ(t)dt = aT − \frac{cq(1 − exp(−Dτ_{0}))}{bD(1 − (1 − p_{2}) exp(−Dτ_{0}))} = − ln(1 − p) > 0,

we have ϕ(τ_{0}) > 0. This implies that B < 0. Hence BC < 0. Thus, from Theorem 2 of [9], the statement follows.

5 Conclusions

In this paper, we have studied the influences of impulsive perturbations on a predatorprey system with the Monod-Haldane functional response. We have found out that there exists a threshold value that plays a key role on discriminating between the stability of the prey-free periodic solution and the permanence of the system via Floquet theory and the comparison theorem. Furthermore, we have shown that the system has a positive periodic solution which is supercritical under some conditions.

Figure 1: The bifurcation diagrams of the system (2) with an initial value (2.5, 4). (a) x is plotted for T over [0, 39]. (b) y is plotted for T over [0, 39].

Figure 2: The bifurcation diagrams of the system (2) with an initial value (2.5, 4). (a) x is plotted for T over [21, 25.7]. (b) y is plotted for T over [21, 25.7].

In order to illustrate the dynamics of the system by a numerical example, we give bifurcation diagrams in Figures 1 and 2 when the parameters are fixed except the period T as follows:

a = 4.0, K = 1.9, b = 0.6, c = 0.75, D = 0.25, m = 0.6, p_{1} = 0.3, p_{2} = 0.001 and q = 1.2.

These figures point out that system (2) has various dynamical behaviors such as quasiperiodic, periodic windows, strange attractors and periodic doubling and halving phenomena etc. (see Figure 2). In this case, we can obtain the critical value T^{∗} ≈ 1.58 suggested in Theorems 3.1 and 3.2. As mentioned in Theorem 4.1, the value T^{∗} plays an important part in the classification for the existence of a positive periodic solution as shown in Figure 1.

Acknowledgements

We would thank the referee for carefully reading the manuscript and for suggesting improvements.

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(Received July 15, 2009)