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Mathematical Problems in Engineering
Volume 2013, Article ID 329592, 11 pages
http://dx.doi.org/10.1155/2013/329592
Research Article

Stability Analysis of a Harvested Prey-Predator Model with Stage Structure and Maturation Delay

1Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110819, China
2State Key Laboratory of Integrated Automation of Process Industry, Northeastern University, Shenyang, Liaoning 110819, China
3Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong

Received 7 February 2013; Accepted 11 April 2013

Academic Editor: Ligang Wu

Copyright © 2013 Chao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A harvested prey-predator model with density-dependent maturation delay and stage structure for prey is proposed, where selective harvest effort on predator population is considered. Conditions which influence positiveness and boundedness of solutions of model system are analytically investigated. Criteria for existence of all equilibria and uniqueness of positive equilibrium are also studied. In order to discuss effects of maturation delay and harvesting on model dynamics, local stability analysis around all equilibria of the proposed model system is discussed due to variation of maturation delay and harvest effort level. Furthermore, global stability of positive equilibrium is investigated by utilizing an iterative technique. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

1. Introduction

In the natural world, many species have a life history that takes them through two stages, juvenile stage and adult stage. Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development, and reproduction) of individuals in a population almost always depend on stage structure. Furthermore, many complex biological phenomena arising in prey-predator ecosystem always depend on the past history of system, and it has been recognized that time delay may have complicated impact on dynamics of prey-predator ecosystem [1]. In the past several decades, there has been an increasing interest in prey-predator model system with stage structure and time delay (see [226] and the references therein).

In the model proposed by Aiello and Freedman [2], stage structure of single population growth with stage structure and time delay representing for maturation of population is considered. Their model predicts a positive steady state as the global attractor, thereby suggesting that stage structure does not generate sustained oscillations frequently observed in single population in the real world. Subsequent work made by other authors [3, 6, 7, 1214] suggests that time delay to adulthood should be state dependent. Generally, boundedness and persistence of solutions of model system may be affected by introduction of time delay into prey-predator system with stage structure [14, 15, 2022, 2426]. Time delay can also cause loss of stability and other complicated dynamical behavior [27]. Especially, there is a well-developed theory of stage-structured models which incorporate time delay into maturity of population [4].

It is well known that harvesting has a strong impact on dynamic evolution of a population; there has been considerable interest in the modeling of harvesting of biological resources [1]. In these models, the harvesting effort is considered to be a dynamic variable; several kinds of harvesting policies are utilized to study the dynamical behavior of the model system. In recent years, there has been growing interest in the study of stage-structured prey-predator system with harvesting. Several prey-predator models with stage structure and harvest effort on predator have been investigated in [2833] and the references therein.

Recently, Huo et al. [24] investigated dynamical behavior and stability of the following stage-structured system with time delay: where , , and represent the density of immature prey population, mature prey population and predator population, at time , respectively; is the intrinsic growth rate of mature prey population, and is the death rate of immature prey population. Constant denotes maturation delay of immature prey population to mature prey population, and the term represents the immature prey population who were born at time and survived at time . denotes the intracompetition rate for mature prey population due to overcrowding phenomenon with mature prey population. is the maximum value of the per capita reduction rate of mature prey population due to predator population, and is the maximum value of the per capita reduction rate of predator population due to mature prey population. measures the extent to which the environment provides protection to mature prey population, and measures the extent to which the environment provides protection to predator population. represents the maximal per capita growth rate of predator population. All the parameters mentioned previously are all positive constants. Furthermore, global stability of positive equilibrium of model system (1) is investigated in [26].

It is well known that the length of time for prey population to maturity is density dependent; that is, maturation time depends on the total population amount of prey population within prey predator ecosystem, and prey population takes less time to reach maturity with depletion of predator population [23, 3436]. Density-dependent maturity of population in prey predator ecosystem is discussed in their work, which reveals that density-dependent effects of the predators’ counterparts to prey defenses and the density dependence effect of each type of predator offense are analogous to the corresponding type of prey defense. Dynamical behavior and stability switch is investigated in [23, 3436]. However, harvest effort on population within prey-predator ecosystem is not considered in [23, 3436].

By assuming maturity delay of prey population is density dependent and predator population is harvested; work done in [24] is extended in this paper, and a harvested prey predator model with density-dependent maturation delay and stage structure for prey population is proposed in the second section of this paper. In the third section of this paper, positiveness and boundedness of solution of the proposed model are studied, and the conditions for existence of equilibria and uniqueness of positive equilibrium are also investigated. Local stability analysis around all equilibria is discussed due to variation of maturation delay as well as harvest effort level. Furthermore, global stability of the positive equilibrium of the proposed model system is studied by utilizing an iterative technique. In the fourth section of this paper, numerical simulations are carried out to show consistency with theoretical analysis. Finally, this paper ends with a conclusion.

2. Model Formulation

Based on the previous analysis, the model proposed by Huo et al. in [24] is extended by incorporating harvest effort on predator population and assuming that maturation delay of prey population is density dependent, and the model can be governed by the following differential equations: The initial conditions for model system (2) take the following form: where , a scalar denotes the harvesting effort to predator population, constant is the catchability coefficient of predator, and the harvesting term follows the catch per unit effort hypothesis [1]. Furthermore, , , , , ,, , and in model system (2) share the same interpretations mentioned in model system (1).

In the following section of this paper, model system (2) is derived under the following hypotheses. (H1) Prey population is divided into two-stage groups, that is, immature and mature. The term represents the immature prey population born at time and survive at time with death rate , which represents transformation term from immature prey to mature prey.(H2) Density-dependent time delay is taken to be an increasing differentiable bounded function of the total population (immature prey, mature prey, and predator population), which satisfies (H3) For the continuity of initial conditions, it is required that where is assumed to be continuous function (for mathematical reason) and nonnegative (for biological reason).(H4) In order to exclude the possibility of immature prey becoming mature prey except by birth, is assumed to be a strictly increasing function of . Otherwise, there are two different times at which the same individual immature prey turns to be mature prey twice at the same instant of time, which is absurd to practical biological interpretations. (For detailed methodology, see [3].)

3. Qualitative Analysis of Model System

In this section, positiveness and boundedness of solution of model system (2) are analytically investigated. Criteria for existence of equilibria and uniqueness of positive equilibrium are also studied. By using differential dynamical system theory and stability theory, local stability analysis around all equilibria of model system is discussed. Furthermore, global stability of the positive equilibrium of the proposed model system is studied by utilizing an iterative technique.

3.1. Positiveness and Boundness of Solutions

Theorem 1. Under hypotheses (H1)–(H4), solutions of model system (2) with given initial conditions are positive for all .

Proof. Assume that there exists . Based on the continuity , it can be computed by evaluating the model system (2) at time :
According to (6) and the initial conditions of model system (2), it is easy to show that . On the other hand, it follows from the definition of that , which is a contradiction. Consequently, for all .
Based on the positiveness of and the third equation of model system (2), it is easy to show that for .
Consider the equation
It is obvious to show that ; that is, is strictly decreasing. By virtue of positiveness of , , it derives that
By solving (7), it gives that
According to (5), it derives that
By substituting in the above equation, it can be obtained that which implies that
According to for all and is an increasing function based on (H4). holds for , and the following inequality can be obtained: since and is strictly decreasing, , . By repeating this argument to include all positive time, it can be shown that for all . Hence, solutions of model system (2) with given initial conditions are positive for all .

Theorem 2. If the hypotheses (H1)–(H4) hold and , all solutions of model system (2) are bounded within a region : where .

Proof. Let , and it is easy to show that based on positiveness of solutions of model system (2). By calculating the derivative of along the solutions, it gives that
By using the standard comparison principle in (15), it derives that
It follows from the third equation of model system (2) that
By utilizing the standard comparison principle in inequality (17) and , it gives that
Consequently, all solutions of model system (2) are bounded within a region : where , .

3.2. Existence of Equilibria and Uniqueness of Positive Equilibrium

The existence of biologically reasonable equilibria of model system (2) is investigated in this subsection. Since the biological interpretation of the positive equilibrium implies that immature prey, mature prey, and predator population all exist, uniqueness of positive equilibrium is also studied.

By simple computation, there are two equilibria denoted by and . The biological interpretations of , are as follows. For , it implies that all population in harvested prey predator ecosystem does not exist. For , it implies that there is not any predation source for predator population. It follows from the previous biological interpretations that population in such ecosystem cannot be maintained at an ideal level for sustainable development, which are not relevant to major investigation in this paper.

Furthermore, there is one or more positive equilibria denoted by . In order to discuss the existence of , it is equivalent to show that the following equations always have at least one positive solution:

It follows from (20) that where and .

Theorem 3 (existence of positive equilibrium). Supposing that hypotheses (H1)–(H4) hold, if , , and , then there exists at least one positive equilibrium .

Proof. Let and be the solution curves of (22) and (23) for , , respectively. The analytical properties of curve and are as follows.
For : by simple computing, it can be found that .
According to (H2) and positiveness of all solutions of model system (2), it is easy to show that , and
For : by differentiating against along , it can be obtained that
It can be shown that , provided that , , and then is strictly decreasing.
Furthermore, according to , , and ,
Based on the above analysis, and intersect at some positive values, which proves the existence of positive equilibrium .

Theorem 4 (uniqueness of positive equilibrium). Supposing that hypotheses (H1)–(H4) hold, if the following inequality holds then there exists a unique positive equilibrium.

Proof. Based on (22) and (23), can be defined as the function of : The positive equilibrium will be unique, provided that for every such otherwise reverse inequality holds.
By differentiating (22) with respect to , it can be obtained that
By differentiating (23) with respect to , it can be obtained that
On the other hand, some expressions about positive equilibrium can be obtained based on (22) and (23),
According to (31), is equivalent to the following inequality:
This completes the proof.

3.3. Local Stability Analysis around Equilibria

Local stability of model system (2) around all equilibria of model system (2) is investigated. Furthermore, stability switch due to variation of maturity delay and harvest effort level is also studied in this subsection.

The characteristic equation of model system (2) about some equilibrium takes the following form: where .

Theorem 5. Local stability analysis of model system (2) around and is as follows: (a)if , then model system is locally stable around , and is a saddle point which is unstable in the -direction and stable in the - plane; (b)if , then model system is locally stable around , and is a saddle point which is unstable in the -direction and stable in the - plane.

Proof. For , (33) reduces to
By solving (34), it can be found that there are two negative eigenvalues and only one positive eigenvalue, provided , which implies that is a saddle point which is unstable in the -direction and stable in the - plane. On the other hand, there are three negative eigenvalues, provided that , which implies that is a stable point.
For , (33) reduces to where . It follows from (35) that there are two negative eigenvalues and only one positive eigenvalue, provided , which implies that is a saddle point which is unstable in the -direction and stable in the - plane. On the other hand, there are three negative eigenvalues, provided that , which implies that is a stable point.

In order to discuss the local stability of model system (2) around the positive equilibrium , (33) reduces towhere , and .

It can be computed that where and ,

In the following part, dynamical behavior of model system (2) around the positive equilibrium is investigated. Furthermore, local stability analysis is discussed due to the variation of maturation delay and harvest effort level. By taking as a bifurcation parameter, conditions for local stability switch are discussed with the increase of from zero.

Case 1 . In the case of , it derives that remains as a constant (zero or a positive constant) for all time based on (H2). In the following part, is denoted as for simplifying. Furthermore, it can be computed that , and () in (37) can be rewritten as follows:

Theorem 6. Supposing that hypotheses (H1)–(H4) hold, if , then model system (2) is stable around the positive equilibrium in the case of .

Proof. When , (37) can be rewritten as follows:
Based on the above analysis, it can be concluded that the roots of (40) have negative real parts by using the Routh-Hurwitz criteria [1]. Consequently, is locally stable in the case of .

When , let be a root of (37), where is positive. Substitute into (37) and separate the real and imaginary parts, and then two transcendental equations can be obtained as follows:

By squaring and adding (41), it can be obtained that where , , , and , () have been defined in (40).

According to the values of () and the Routh-Hurwitz criteria [1], a simple assumption of the existence of a positive root for (42) is .

If holds, then (42) has a positive root , and (37) has a pair of purely imaginary roots of the form . Consequently, it can be obtained by eliminating from (41): The corresponding to is as follows: . By virtue of Butler’s lemma [37], it can be concluded that the positive equilibrium remains locally stable for , as .

Case 2 . In the case of , local stability of model system (2) around the positive equilibrium can change only if there exists at least one root of (37) such that .
Let be one such root, where is positive. Substitute into (37) and separate the real and imaginary parts, and then two transcendental equations can be obtained as follows:
By squaring and adding (45), it can be obtained that where ,, , and , () have been defined in (37).
According to the values of () and the Routh-Hurwitz criteria [1], a simple assumption of the existence of a positive root for (42) is , which derives that
If the above inequality holds, then model system (2) is unstable around the positive equilibrium in the case of .

3.4. Global Stability Analysis of Positive Equilibrium

In this section, global stability of the positive equilibrium is discussed by using an iterative technique in the case of .

Lemma 7 (see [29]). Consider the following equation: where , , , and are positive constants, and for ; it follows that (i)If , then ; (ii)If , then .

Theorem 8. Supposing that hypotheses (H1)–(H4) and hold, if the following inequalities hold then the positive equilibrium is globally asymptotically stable in the case of .

Proof. In the case of , it derives that remains as a constant (zero or a positive constant) for all time based on (H2). In the following part, is denoted as for simplifying. Let
In the following, we will claim that , .
It follows from Theorem 2 that ( has been defined in Theorem 2). From model system (2),
By standard comparison argument, it derives that and then for sufficiently small , there exists a such that if , . Based on Theorem 1, , it can be obtained that for ,
Consider the following auxiliary equation:
Under the condition , it follows from Lemma 7 that
Hence, . For sufficiently small , there exits such that if , then .
We derive from the model system (2) that for ,
A standard comparison argument shows that
Hence, for sufficiently small , there is a satisfying if , then . Consequently, for ,
Consider the following auxiliary equation:
It follows from Lemma 7 that
Hence, . For sufficiently small , there exists a satisfying that if , then . For , it gives that
By standard comparison argument, it derives that
Hence, for sufficiently small , there exists satisfying that if , , the for
Consider the following auxiliary equation:
By using Lemma 7, it can be obtained that
Since it is true for any sufficiently small , . Therefore, there exists such that if , then .
It follows from model system (2) that for ,
By using standard comparison argument, it derives that
Since this is true for any sufficiently small , . Consequently, there exists satisfying if , then .
It follows from model system (2) that for ,
Consider the following auxiliary equation,
By using Lemma 7, it derives that
Continuing the above process, four sequences , , , , , are obtained which take the following form
It is easy to show that
By virtue of (71), it derives that
Furthermore,
If the following inequalities hold then , which implies that is monotonically decreasing. Hence, it can be shown that limitation of sequence exists. Taking , it follows from (73) that
By using (71) and (76), it can be shown that
According to the definition of , , , and , it derives that
Hence,
Based on (5), it derives that
By using L’Hospital’s rule, it derives that
According to (20), it is easy to show that
This completes the proof.

4. Numerical Simulation

With the help of MATLAB, numerical simulations are provided to understand the theoretical results which have been established in the previous sections of this paper. In order to facilitate the numerical simulation, it is assumed that takes the following form [23]: where satisfying . Based on Theorem 1, it follows from simple computation that which implies that (H2) holds.

Values of parameters are taken from [24] which are used in Example 1 of [24] and set in appropriate units. , , , , , , , , , and . According to the given values of parameters, it follows from Theorems 3 and 4 that there exists a unique positive equilibrium . Furthermore, it can be verified that is globally attractive based on Theorem 8. Responses of model system (2) are indicated in Figure 1, and the phase portrait of model system (2) with different initial values is plotted in Figure 2.

329592.fig.001
Figure 1: Dynamical responses of model system (2).
329592.fig.002
Figure 2: Phase portrait of model system (2) with different initial values.

5. Conclusion

In this paper, a harvested prey predator model is proposed to investigate the effects of density-dependent maturation delay and harvest effort on the dynamics. Conditions which influence positiveness and boundedness of solutions of model system are obtained in Theorems 1 and 2, respectively. Existence of all equilibria of model system and uniqueness of the positive equilibrium are studied in Theorems 3 and 4, respectively. Biological interpretations of the positive equilibrium mean immature prey, mature prey, predator and harvest effort on predator population all exist in the harvested ecosystem. Consequently, we mainly concentrate on dynamical analysis around positive equilibrium in this paper. Local stability analysis in Theorem 6 reveals that local stability of the positive equilibrium loses due to variation of maturation delay and harvest effort level. Furthermore, global stability of the positive equilibrium is discussed by utilizing an iterative technique in Theorem 8, which is utilized to investigate the coexistence and interaction mechanism of harvested prey-predator ecosystem.

Compared with the work done in [24] and the related work in [26], maturation delay for prey population in this paper relates to the density of all population within the harvested ecosystem, which accurately reflects the practical phenomena in the real world [23, 3436]. Furthermore, it should be noted that dynamics of prey predator model with density-dependent delay for predator population is investigated in [23], while dynamics of harvest effort on population within ecosystem is not considered. Compared with the work done in [23], harvest effort on predator population is introduced, and the effect of harvesting on model dynamics is also investigated in this paper. With the rapid development of commercial harvesting on prey predator ecosystem in the real world, the introduction of harvest effort and related qualitative analysis makes the work done in this paper have some new and positive feature.

Acknowledgments

The authors gratefully acknowledge editors and anonymous reviewers' comments and patient work. This work is supported by National Natural Science Foundation of China, grant No. 61104003, grant No. 61273008 and grant No. 61104093. Research Foundation for Doctoral Program of Higher Education of Education Ministry, grant No. 20110042120016. Hebei Province Natural Science Foundation, grant No. F2011501023. Fundamental Research Funds for the Central Universities, grant No. N120423009. Research Foundation for Science and Technology Pillar Program of Northeastern University at Qinhuangdao, grant No. XNK201301. This work is supported by State Key Laboratory of Integrated Automation of Process Industry, Northeastern University, supported by Hong Kong Admission Scheme for Mainland Talents and Professionals, Hong Kong Special Administrative Region.

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