Abstract
We proposed and analyzed a mathematical model dealing with two species of prey-predator system. It is assumed that the prey is a stage structure population consisting of two compartments known as immature prey and mature prey. It has a refuge capability as a defensive property against the predation. The existence, uniqueness, and boundedness of the solution of the proposed model are discussed. All the feasible equilibrium points are determined. The local and global stability analysis of them are investigated. The occurrence of local bifurcation (such as saddle node, transcritical, and pitchfork) near each of the equilibrium points is studied. Finally, numerical simulations are given to support the analytic results.
1. Introduction
The development of the qualitative analysis of ordinary differential equations is deriving to study many problems in mathematical biology. The modeling for the population dynamics of a prey-predator system is one of the important and interesting goals in mathematical biology, which has received wide attention by several authors [1–6]. In the natural world many kinds of prey and predator species have a life history that is composed of at least two stages: immature and mature, and each stage has different behavioral properties. So, some works of stage structure prey-predator models have been provided in a good number of papers in the literatures [7–12]. Zhang et al. in [9] and Cui and Takeuchi in [11] proposed two mathematical models of prey stage structure; in these models the predator species consumes exclusively the immature prey. Indeed, there are many factors that impact the dynamics of prey-predator interactions such as disease, harvesting, prey refuge, delay, and many other factors. Several prey species have gone to extinction, and this extinction must be caused by external effects such as overutilization, overpredation, and environmental factors (pollution, famine). Prey may avoid becoming attacked by predators either by protecting themselves or by living in a refuge where it will be out of sight of predators. Some theoretical and empirical studies have shown and tested the effects of prey refuges and making an opinion that the refuges used by prey have a stabilizing influence on prey-predator interactions; also prey extinction can be prevented by the addition of refuges; for instance, we can refer to [13–23]. Therefore, it is important and worthwhile to study the effects of a refuge on the prey population with prey stage structure. Consequently, in this paper, we proposed and analyzed the prey-predator model involving a stage structure in prey population together with a preys refuge property as a defensive property against the predation.
2. Mathematical Model
In this section a prey-predator model with a refuge-stage structure prey population is proposed for study. Let represent the population size of the immature prey at time ; represents the population size of the mature prey at time , while denotes the population size of the predator species at time . Therefore in order to describe the dynamics of this model mathematically the following hypotheses are adopted:(1)The immature prey grows exponentially depending completely on its parents with growth rate . There is an intraspecific competition between their individuals with intraspecific competition rate . The immature prey individual becomes mature with grownup rate and faces natural death with a rate .(2)There is an intraspecific competition between the individuals of mature prey population with intraspecific competition rate . Further the mature prey species faces natural death rate too with a rate .(3)The environment provides partial protection of prey species against the predation with a refuge rate ; therefore there is of prey species available for predation.(4)There is an intraspecific competition between the individuals of predator population with intraspecific competition rate . Further the predator species faces natural death rate too with a rate .The predator consumes the prey in both the compartments according to the mass action law represented by Lotka–Volterra type of functional response with conversion rates and for immature and mature prey, respectively. Consequently, the dynamics of this model can be represented mathematically with the following set of differential equations.with initial conditions, , , and .
In order to simplify the analysis of system (1) the number of parameters in system (1) is reduced using the following dimensionless variables in system (1):Accordingly, the dimensionless form of system (1) can be written aswhere the dimensionless parameters are given byAccording to the equations given in system (3), all the interaction functions are continuous and have a continuous partial derivatives. Therefore they are Lipschitzain and hence the solution of system (3) exists and is unique. Moreover the solution of system (3) is bounded as shown in the following theorem.
Theorem 1. All the solutions of system (3) that initiate in the positive octant are uniformly bounded.
Proof. Let be solutions of system (3) with initial conditions, , , and Then by differentiation with respect to we gethere and Now by using comparison theorem, we getThus for we obtain . Hence, all solutions of system (3) in are uniformly bounded and therefore we have finished the proof.
3. Local Stability Analysis
It is observed that system (3) has at most three biologically feasible equilibrium points; namely, , The existence conditions for each of these equilibrium points are discussed below:(1)The trivial equilibrium points exist always.(2)The predator free equilibrium point is denoted by , where while is a positive root of the following third-order polynomial here, , , , and exist uniquely in the interior of -plane if and only if the following condition holds:(3)The interior (positive) equilibrium point is given by , where while is a positive root of the following third-order polynomial: here, Clearly, (11) has a unique positive root represented by if the following set of conditions hold: Therefore, exists uniquely in int. if in addition to condition (13) the following conditions are satisfied.Now to study the local stability of these equilibrium points, the Jacobian matrix for the system (3) at any point is determined asThus, system (3) has the following Jacobian matrix near .Then the characteristic equation of is given byClearly, all roots of (17) have negative real parts if and only if the following condition holds:So, is locally asymptotically stable under condition (18) and saddle point otherwise. Therefore, is locally asymptotically stable whenever does not exist and unstable whenever exists.
The Jacobian matrix of system (3) around the equilibrium point reduced toThen the characteristic equation of is written byStraightforward computation shows that all roots of (20) have negative real part provided thatSo, is locally asymptotically stable if the above two conditions hold.
Finally the Jacobian matrix of system (3) around the interior equilibrium point is writtenhereHence, the characteristic equation of becomeswithConsequently, can be written as Since , then according to Routh-Hurwitz criterion is locally asymptotically stable if and only if and . According to the form of and the signs of Jacobian elements the last four terms are positive, while the first term will be positive under the sufficient condition (28) below. However becomes positive if and only if in addition to condition (28) the second sufficient condition given by (29) holds.Therefore under these two sufficient conditions is locally asymptotically stable.
4. Global Stability
In this section the global stability for the equilibrium points of system (3) is investigated by using the Lyapunov method as shown in the following theorems.
Theorem 2. Assume that the vanishing equilibrium point is locally asymptotically stable; then it is globally asymptotically stable in if and only if the following condition holds:
Proof. Consider the following positive definite real valued function:Straightforward computation shows that the derivative of with respect to is given byTherefore, by using condition (30), we obtain which is negative definite in , and then is a Lyapunov function with respect to . Hence is globally asymptotically stable in and the proof is complete.
Theorem 3. Assume that the predator free equilibrium point is locally asymptotically stable; then it is globally asymptotically stable in if the following condition holds:
Proof. Consider the following positive definite real valued function:Straightforward computation shows that the derivative of with respect to is given byNow using condition (33) gives us thatClearly is negative definite due to local stability condition (21). Hence is a Lyapunov function with respect to , and then is globally asymptotically stable, which completes the proof.
Theorem 4. Assume that the interior equilibrium point is locally asymptotically stable in ; then it is globally asymptotically stable if and only if the following condition holds:
Proof. Consider the following positive definite real valued function around :Our computation for the derivative of with respect to gives thatNow by using the condition (37) we obtain thatAccording to the above inequality we have which is negative definite; therefore, is globally asymptotically stable in and hence the proof is complete.
5. Local Bifurcation
In this section the local bifurcation near the equilibrium points of system (3) is investigated using Sotomayor’s theorem for local bifurcation [24]. It is well known that the existence of nonhyperbolic equilibrium point is a necessary but not sufficient condition for bifurcation to occur. Now rewrite system (3) in the formwhereThen according to Jacobian matrix of system (3) given in (15), it is simple to verify that for any nonzero vector we haveConsequently, we obtain thatand thereforeThus system (3) has no pitchfork bifurcation due to (45). Moreover, the local bifurcation near the equilibrium points is investigated in the following theorems:
Theorem 5. System (3) undergoes a transcritical bifurcation near the vanishing equilibrium point, but saddle node bifurcation cannot occur, when the parameter passes through the bifurcation value .
Proof. According to the Jacobian matrix given by (16), system (3) at the equilibrium point with has zero eigenvalue, say , and the Jacobian matrix becomesNow let be the eigenvector corresponding to the eigenvalue . Thus gives , where represents any nonzero real number. Also, let represents the eigenvector corresponding to the eigenvalue of . Hence gives that , where denotes any nonzero real number. Now, sincethus , which gives . So, according to Sotomayor’s theorem for local bifurcation, system (3) has no saddle node bifurcation at . Also, sincethen,Moreover, by substituting , and in (44) we get thatHence, it is obtain thatThus, according to Sotomayor’s theorem system (3) has a transcritical bifurcation at as the parameter passes through the value ; thus the proof is complete.
Theorem 6. Assume that condition (22) holds; then system (3) undergoes a transcritical bifurcation near the predator free equilibrium point , but saddle node bifurcation cannot occur, when the parameter passes through the bifurcation value .
Proof. According to the Jacobian matrix given by (19), system (3) at the equilibrium point with has zero eigenvalue, say , and the Jacobian matrix becomeswhere , with . Now let, be the eigenvector corresponding to the eigenvalue . Thus gives , where and are negative according to the sign of the Jacobian elements and represents any nonzero real numbers. Also, let represent the eigenvector corresponding to eigenvalue of . Hence gives that , where stands for any nonzero real numbers. Now, sincethus , which gives . So, according to Sotomayor’s theorem for local bifurcation, system (3) has no saddle node bifurcation at . Also, sincethen, we can haveMoreover, substituting , and in (44) givesHence, it is obtained thatThus, according to Sotomayor’s theorem system (3) has a saddle node bifurcation at as the parameter passes through the value ; thus the proof is complete.
Theorem 7. Assume that condition (21) holds; then system (3) undergoes a saddle node bifurcation near the predator free equilibrium point when the parameter passes through the bifurcation value .
Proof. According to the Jacobian matrix given by (19), system (3) at the equilibrium point with has zero eigenvalue, say , and the Jacobian matrix becomeswhere , with . Now let be the eigenvector corresponding to the eigenvalue . Thus gives , where represents any nonzero real numbers. Also, let represent the eigenvector corresponding to eigenvalue of . Hence gives that , where is negative due to the sign of the Jacobian elements and denotes any nonzero real numbers. Now, sincethus ; hence . So, according to Sotomayor’s theorem for local bifurcation the first condition of saddle node bifurcation is satisfied in system (3) at . Moreover, substituting , and in (44) givesHence, it is obtained thatThus, according to Sotomayor’s theorem system (3) has a transcritical bifurcation at as the parameter passes through the value ; thus the proof is complete.
Theorem 8. Assume thatThen system (3) undergoes a saddle node bifurcation near the interior equilibrium point , as the parameter passes through the bifurcation value , where and are given in the proof.
Proof. According to the determinant of the Jacobian matrix given by in (25), condition (62) represents a necessary condition to have nonpositive determinant for . Now rewrite the form of the determinant as follows:Here and . Obviously, is positive always, while is positive under the condition (63). Thus it is easy to verify that and hence has zero eigenvalue, say , as passes through the value , which means that becomes a nonhyperbolic point. Let now the Jacobian matrix of system (3) at with be given bywhere , and .
Let be the eigenvector corresponding to the eigenvalue . Thus gives , where and are positive due to the Jacobian elements and represents any nonzero real number. Also, let represent the eigenvector corresponding to eigenvalue of . Hence gives that , where and are negative due to the Jacobian elements and denotes any nonzero real numbers. Now, sincethus , which gives . Consequently the first condition of saddle node bifurcation is satisfied. Moreover, by substituting , and in (44) we get thatHence, it is obtained thatSo, according to Sotomayor’s theorem, system (3) has a saddle node bifurcation as passes through the value and hence the proof is complete.
6. Numerical Simulations
In this section, the global dynamics of system (3) is investigated numerically. The objectives first confirm our obtained analytical results and second specify the control set of parameters that control the dynamics of the system. Consequently, system (3) is solved numerically using the following biologically feasible set of hypothetical parameters with different sets of initial points and then the resulting trajectories are drawn in the form of phase portrait and time series figures.
Clearly, Figure 1 shows the asymptotic approach of the solutions, which started from different initial points to a positive equilibrium point (0.46, 0.15, 0.42), for the data given by (69). This confirms our obtained result regarding the existence of globally asymptotically stable positive point of system (3) provided that certain conditions hold.
Now in order to discuss the effect of the parameters values of system (3) on the dynamical behavior of the system, the system is solved numerically for the data given in (69) with varying one parameter each time. It is observed that varying parameters values , and have no qualitative effect on the dynamical behavior of system (3) and the system still approaches to a positive equilibrium point. On the other hand, when decreases in the range keeping other parameters fixed as given in (69) the dynamical behavior of system (3) approaches asymptotically to the vanished equilibrium point as shown in the typical figure given by Figure 2. Similar observations have been obtained on the behavior of system (3) in case of increasing the parameter in the range or decreasing the parameter in the range , with keeping other parameters fixed as given in (69), and then the solution of system (3) is depicted in Figures 3 and 4, respectively. Finally, for the parameters and with other parameters fixed as given in (69), the solution of system (3) approaches asymptotically to the predator free equilibrium point as shown in the Figure 5.
7. Discussion
In this paper, a model that describes the prey-predator system having a refuge and stage structure properties in the prey population has been proposed and studied analytically as well as numerically. Sufficient conditions which ensure the stability of equilibria and the existence of local bifurcation are obtained. The effect of each parameter on the dynamical behavior of system (3) is studied numerically and the trajectories of the system are drowned in the typical figures. According to these figures, which represent the solution of system (3) for the data given by (69), the following conclusions are obtained.(1)System (3) has no periodic dynamics rather than the fact that the system approaches asymptotically to one of their equilibrium points depending on the set of parameter data and the stability conditions that are satisfied.(2)Although the position of the positive equilibrium point in the interior of changed as varying in the parameters values , and , there is no qualitative change in the dynamical behavior of system (3) and the system still approaches to a positive equilibrium point. Accordingly adding the refuge factor which is included implicitly in these parameters plays a vital role in the stabilizing of the system at the positive equilibrium point.(3)Decreasing in the value of growth rate of immature prey or in the value of conversion rate from immature prey to mature prey keeping the rest of parameter as in (69) leads to destabilizing of the positive equilibrium point and the system approaches asymptotically to the vanishing equilibrium point, which means losing the persistence of system (3).(4)Increasing in the value of grownup rate of immature prey keeping the rest of parameter as in (69) leads to destabilizing of the positive equilibrium point and the system approaches asymptotically to the vanishing equilibrium point too, which means losing the persistence of system (3).(5)Finally, for the data given by (69), increasing intraspecific competition of immature prey and natural death rate of the predator leads to destabilizing of the positive equilibrium point and the solution approaches instead asymptotically to the predator free equilibrium point, which confirm our obtained analytical results represented by conditions (21)-(22).
Competing Interests
The authors declare that they have no competing interests.