Stability Analysis of a Ratio-Dependent Predator-Prey Model

where x(t) and y(t) are the densities of the prey and predator population at time t, respectively.+e functionf(x) represents the growth of the prey population rate, g(y) represents the growth rate of predator population, and p(x) represents the functional response function of predator population to prey population. In [1], Xu et al. used the function p(x) � x2/(x2 + my2) as the functional response function of predator population to prey population. +e time delay due to the gestation of the predator is discussed in [1]. It is noted that in model (1), each individual’s prey admits the same risk to be attacked by predators and each individual predator admits the same ability to feed on prey. +is assumption seems not to be realistic for many animals. In natural world, there are many species whose individuals pass through an immature stage. Stage structure is a natural phenomenon and represents, for example, the division of a population into immature andmature individuals. In the last two decades, stage-structured models have received great attention [3–7, 9]. Based on above discussion, we study the following predator-prey model:


Introduction
Recently, the predator-prey models have been studied by many authors [1][2][3][4][5][6][7][8]. In general, a predator-prey model has the following forms: where x(t) and y(t) are the densities of the prey and predator population at time t, respectively. e function f(x) represents the growth of the prey population rate, g(y) represents the growth rate of predator population, and p(x) represents the functional response function of predator population to prey population. In [1], Xu et al. used the function p(x) � x 2 /(x 2 + my 2 ) as the functional response function of predator population to prey population. e time delay due to the gestation of the predator is discussed in [1]. It is noted that in model (1), each individual's prey admits the same risk to be attacked by predators and each individual predator admits the same ability to feed on prey.
is assumption seems not to be realistic for many animals. In natural world, there are many species whose individuals pass through an immature stage. Stage structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals. In the last two decades, stage-structured models have received great attention [3][4][5][6][7]9].
Based on above discussion, we study the following predator-prey model: where x 1 (t) and x 2 (t) are the densities of the immature and mature prey at time t and y 1 (t) and y 2 (t) are the densities of the immature and mature predators at time t. In model (2), all parameters are positive constants. τ ≥ 0 is the time delay due to the gestation of the predator. x 2 /(x 2 + my 2 ) is the ratio-dependent functional response. Model (2) is of the following initial conditions: e organization of this study is as follows. In Section 2, we discuss the local stability of the nonnegative boundary equilibrium and the positive equilibrium of models (2) and (3). e existence of a Hopf bifurcation for models (2) and (3) at the positive equilibrium is also established. Sufficient conditions are derived for the global stability of the nonnegative boundary equilibrium and positive equilibrium of models (2) and (3) in Section 3, respectively.

Local Stability and Hopf Bifurcation
In this section, by analyzing the corresponding characteristic equations, we study the local stability of each of nonnegative equilibria and the existence of a Hopf bifurcation at the positive equilibrium of models (2) and (3).

Global Stability
In this section, by using an iteration technique, we discuss the global stability of the nonnegative equilibria E 1 and E + of models (2) and (3), respectively.

Theorem 2. Let
hold; then, the nonnegative boundary equilibrium E 1 of model (2) is globally stable.
Proof. It follows from the positive solution of model (2), and we can obtain By Lemma 2.2 of [5] and comparison, we have erefore, there is a positive number t 1 , for sufficiently small positive number ε, such that as t > t 1 , x 1 (t) ≤ x 1 ′ + ε. Hence, for t > t 1 + τ, we derive that By Lemma 2.2 of [5] and comparison, we can obtain  erefore, there is a positive number For t > t 2 , we derive from model (2) that By Lemma 2.2 of [5] and comparison, we have (23) By model (2), it follows that By Lemma 2.4 of [3] and comparison, we obtain that which together with (19) and (21) yields Hence, the equilibrium E 1 (x 1 ′ , x 2 ′ , 0, 0) of model (2) is globally stable. , L y i � liminf t⟶+∞ y i (t), (i � 1, 2).

(28)
By the first two equations of model (2), we can obtain that By Lemma 2.2 of [5] and comparison, we have So, for sufficiently small positive number ε, there exists a positive number t 1 , such that if t > t 1 , then For t > t 1 + τ, by the last two equations of model (2), we get By Lemma 2.2 of [5] and comparison, we obtain (32) erefore, for sufficiently small positive number ε, there is For t > t 2 , by the first two equations of model (2), we have (34) By Lemma 2.4 of [3] and comparison, we derive that Hence, for sufficiently small positive number ε, there is For t > t 3 + τ, it follows from the last two equations of model (2) that By Lemma 2.4 of [3] and comparison, we can obtain erefore, for sufficiently small positive number ε, there is a positive number t 4 ≥ t 3 + τ, such that if t > t 4 , y 2 (t) ≥ N y 2 1 − ε. In this case, by the first two equations of model (2), we have Journal of Mathematics erefore, for sufficiently small positive number ε, there is t 5 ≥ t 4 , such that if t > t 5 , From the last two equations of model (2), we obtain that for t > t 5 + τ, By Lemma 2.2 of [5] and comparison, if a 2 r 2 > d 4 (r 2 + d 3 ) holds, we have Hence, for ε > 0 sufficiently small, there is a T 6 ≥ T 5 + τ, such that if t > T 6 , y 2 (t) ≤ M y 2 2 + ε. Again, for sufficiently small positive number ε and t > t 6 , by the first two equations of model (2), we have (43) So, there is a positive number t 7 ≥ t 6 , for t > t 7 , For sufficiently small positive number ε and t > 7 + τ, from the last two equations of model (2), we can derive