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Dynamics of a Predator-Prey System Concerning Biological and Chemical Controls
Journal of Inequalities and Applications volume 2010, Article number: 598495 (2010)
Abstract
We investigate an impulsive predator-prey system with Monod-Haldane type functional response and control strategies, especially, biological and chemical controls. Conditions for the stability of the prey-free positive periodic solution and for the permanence of the system are established via the Floquet theory and comparison theorem. Numerical examples are also illustrated to substantiate mathematical results and to show that the system could give birth to various kinds of dynamical behaviors including periodic doubling, and chaotic attractor. Finally, in discussion section, we consider the dynamic behaviors of the system when the growth rate of the prey varies according to seasonal effects.
1. Introduction
In recent years controlling insects and other arthropods has become an increasingly complex issue. There are many ways that can be used to help control the population of insect pests. Integrated Pest Management (IPM) is a pest control strategy that uses an array of complementary methods: natural predators and parasites, pest-resistant varieties, cultural practices, biological controls, various physical techniques, and the strategic use of pesticides.
Chemical control is one of simple methods for pest control. Pesticides are often useful because they quickly kill a significant portion of a pest population. However, there are many deleterious effects associated with the use of chemicals that need to be reduced or eliminated. These include human illness associated with pesticide applications, insect resistance to insecticides, contamination of soil and water, and diminution of biodiversity. As a result, it is required that we should combine pesticide efficacy tests with other ways of control. Another important way to control pest populations is biological control. It is defined as the reduction of pest populations by natural enemies and typically involves an active human role. Natural enemies of insect pests, also known as biological control agents, include predators, parasites, and pathogens. Virtually all pests have some natural enemies, and the key to successful pest control is to identify the pest and its natural enemies and release them at fixed times for pest control. Biological control can be an important component of Integrated Pest Management (IPM) programs. Such different pest control tactics should work together rather than against each other to accomplish an IPM program successfully [1, 2].
On the other hand, the relationship between pest and natural enemy can be expressed a predator(natural enemy)-prey(pest) system mathematically as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ1_HTML.gif)
where and
represent the population density of the prey and the predator at time
, respectively. Usually,
is called the carrying capacity of the prey. The constant
is called intrinsic growth rate of the prey. The constants
,
are the conversion rate and the death rate of the predator, respectively. The function
is the functional response of the predator which means prey eaten per predator per unit of time. Many scholars have studied such predator-prey systems with functional response, such as Holling-type [3–5], Beddington-type [6–9], and Ivlev-type [10–12]. One of well-known function response is of Monod-Haldane type [4, 5, 13]. The predator-prey system with Monod-Haldane type is described by the following differential equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ2_HTML.gif)
Therefore, to accomplish the aims discussed above, we need to consider impulsive differential equation as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ3_HTML.gif)
where the parameters and
are the periods of the impulsive immigration or stock of the predator,
present the fraction of the prey which dies due to the harvesting or pesticides and so forth, and
is the size of immigration or stock of the predator.
In fact, impulsive control methods can be found in almost every field of applied sciences. The theoretical investigation and its application analysis can be found in Bainov and Simeonov [14], Lakshmikantham et al. [15]. Moreover, the impulsive differential equations dealing with biological population dynamics are literate in [16–21]. In particular, Zhang et al. [20] studied the system (1.3) without chemical control. That is, . They investigated the abundance of complex dynamics of the system (1.3) theoretically and numerically.
The main purpose of this paper is to investigate the dynamics of the system (1.3). In Section 3, we study qualitative properties of the system (1.3). In fact, we show the local stability of the prey-free periodic solution under some conditions and give a sufficient condition for the permanence of the system (1.3) by applying the Floquet theory. In Section 4 we numerically investigate the system (1.3) to figure out the influences of impulsive perturbations on inherent oscillation. Finally, in Section 5, we consider the dynamic behaviors of the system when the growth rate of the prey varies according to seasonal effects.
2. Basic Definitions and Lemmas
Before stating our main results, firstly, we give some notations, definitions and lemmas which will be useful for our main results.
Let ,
and
. Denote
as the set of all of nonnegative integers and
as the right hand of the system (1.3). Let
, then
is said to be in a class
if
(1)is continuous in
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ4_HTML.gif)
and exists for each
;
(2) is locally Lipschitzian in
.
Definition 2.1.
Let and
. The upper right derivative of
with respect to the impulsive differential system (1.3) is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ5_HTML.gif)
It is from [15] that the smoothness properties of guarantee the global existence and uniqueness of solutions to the system (1.3).
We will use a comparison inequality of impulsive differential equations. Suppose that satisfies the following hypotheses:
-
(H)
is continuous on
and the limits
,
exist and are finite for
and
.
Lemma 2.2 (see [15]).
Suppose that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ6_HTML.gif)
where satisfies
and
are nondecreasing for all
. Let
be the maximal solution for the impulsive Cauchy problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ7_HTML.gif)
defined on . Then
implies that
, where
is any solution of (2.3).
Similar result can be obtained when all conditions of the inequalities in the Lemma 2.2 are reversed. Note that if we have some smoothness conditions of to guarantee the existence and uniqueness of the solutions for (2.4), then
is exactly the unique solution of (2.4).
From Lemma 2.2, it is easily proven that the following lemma holds.
Lemma 2.3.
Let be a solution of the system (1.3). Then one has the following:
(1)if then
for all
;
(2)if then
for all
.
It follows from Lemma 2.3 that the positive quadrant is an invariant region of the system (1.3).
Even if the Floquet theory is well known, we would like to mention the theory to study the stability of the prey-free periodic solution as a solution of the system (1.3). For this, we present the Floquet theory for the linear -periodic impulsive equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ8_HTML.gif)
Then we introduce the following conditions.
()   and
, where
is a set of all piecewise continuous matrix functions which is left continuous at
, and
is a set of all
matrices.
()  ,
.
()   There exists a such that
.
Let be a fundamental matrix of (2.5), then there exists a unique nonsingular matrix
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ9_HTML.gif)
By equality (2.6) there corresponds to the fundamental matrix and the constant matrix
which we call the monodromy matrix of (2.5) (corresponding to the fundamental matrix of
). All monodromy matrices of (2.5) are similar and have the same eigenvalues. The eigenvalues
of the monodromy matrices are called the Floquet multipliers of (2.5).
Lemma 2.4 (Floquet theory [14]).
Let conditions (H1)–(H3) hold. Then the linear -periodic impulsive equation (2.5) is
(1)stale if and only if all multipliers of (2.5) satisfy the inequality
, and moreover, to those
for which
, there correspond simple elementary divisors;
(2)asymptotically stable if and only if all multipliers of (2.5) satisfy the inequality
;
(3)unstable if for some
.
3. Mathematical Analysis
In this section, we have focused on two main subjects, one is about the extinction of the prey and the other is about the coexistence of the prey and the predator. For the extinction, we have found out a condition that the population of the prey goes to zero as time goes by via the study of the stability of a prey-free periodic solution. For example, if the prey is regarded as a pest, it is important to figure out when the population of the prey dies out. For the reason, it is necessary to consider the stability of the prey-free periodic solution. On the other hand, for the coexistence, we have investigated that the populations of the prey and the predator become positive and finite under certain conditions.
3.1. Stability for a Prey-Free Periodic Solution
First of all, in order to study the extinction of the prey, the existence of a prey-free solution to the system (1.3) should be guaranteed. For the reason, we give some basic properties of the following impulsive differential equation which comes from the system (1.3) by setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ10_HTML.gif)
The system (3.1) is a periodically forced linear system; it is easy to obtain from elementary calculations that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_IEq90_HTML.gif)
, is a positive periodic solution of (3.1). Moreover, we can figure out that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ12_HTML.gif)
is the solution of (3.1). From (3.2) and (3.3), the following results can be easily obtained without the proof.
Lemma 3.1.
For every solution and every positive periodic solution
of the system (3.1), it follows that
tend to
as
. Thus, the complete expression for the prey-free periodic solution of the system (1.3) is obtained
.
Now, in the next theorem, the stability of the periodic solution is investigated.
Theorem 3.2.
Let be any solution of the system (1.3). Then the prey-free periodic solution
is locally asymptotically stable if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ13_HTML.gif)
Proof.
The local stability of the periodic solution of the system (1.3) may be determined by considering the behavior of small amplitude perturbations of the solution. Define
. Then they may be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ14_HTML.gif)
where satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ15_HTML.gif)
and , the identity matrix. So the fundamental solution matrix is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ16_HTML.gif)
The resetting impulsive condition of the system (1.3) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ17_HTML.gif)
Note that all eigenvalues of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ18_HTML.gif)
are and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ19_HTML.gif)
the condition is equivalent to the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ20_HTML.gif)
According to Lemma 2.4, is locally stable.
Remark 3.3.
-
(1)
It follows from Theorem 3.2 that the population of the prey could be controlled by using chemical or biological control parameters,
,
,
if the other parameters are fixed. (2) Figure 2 illustrates this phenomenon.
3.2. Permanence
It might be difficult to find out a necessary condition for the stability of the prey-free periodic solution . Due to this fact, it is natural to have a question what a condition that makes all species coexist is. Before answering the question, first of all, we introduce a definition which keeps the concept of coexistence of the prey and the predator.
Definition 3.4.
The system (1.3) is permanent if there exists such that, for any solution
of the system (1.3) with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ21_HTML.gif)
From a biological point of view, the populations of the prey and the predator in the system (1.3) cannot increase up to infinity due to restriction of resources. To show this phenomenon for the system (1.3) mathematically, we prove that all solutions to the system (1.3) are uniformly ultimately bounded in the next proposition.
Proposition 3.5.
There is an such that
for all
large enough, where
is a solution of the system (1.3).
Proof.
Let be a solution of the system (1.3) and let
. Then
, if
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ22_HTML.gif)
When ,
and when
,
. Clearly, the right hand of (3.13), is bounded when
. So we can choose
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ23_HTML.gif)
By Lemma 2.2, we can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ24_HTML.gif)
for . Therefore,
is bounded by a constant for sufficiently large
. Hence there is an
such that
for a solution
with all
large enough.
Thanks to Proposition 3.5, we have only to prove the existence of a positive lower bound for the populations of the prey and the predator to justify the system is permanent.
Theorem 3.6.
The system (1.3) is permanent if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ25_HTML.gif)
Proof.
Suppose is any solution of the system (1.3) with
. From Proposition 3.5, we may assume that
,
,
and
. Let
,
. So, it is easily induced from Lemma 3.1 that
for all
large enough. Now we shall find an
such that
for all
large enough. We will do this in the following two steps.
Step 1.
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ26_HTML.gif)
we can choose ,
small enough such that
and
.  Suppose that
for all
. Then we get
from above assumptions. By Lemma 2.2, we have
and
,
, where
is the solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ28_HTML.gif)
Then there exists such that
for
. So we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ29_HTML.gif)
for . Let
and
. Integrating (3.20) on
,
, we have
. Then we have
as
which is a contradiction. Hence there exists a
such that
.
Step 2.
If for all
, then we are done. If not, we may let
. Then
for
and, by the continuity of
, we have
. In this step, we have only to consider two possible cases.
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_IEq186_HTML.gif)
for some . Then
. Select
such that
and
, where
. Let
. In this case we will show that there exists
such that
. Otherwise, by (3.18) with
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ30_HTML.gif)
and . So we get
and
for
. Also we get to know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ31_HTML.gif)
for . As in Step 1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ32_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ33_HTML.gif)
for . Integrating (3.24) on
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ34_HTML.gif)
Thus which is a contradiction. Now, let
=
. Then
for
and
. So, we have, for
,
.
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_IEq214_HTML.gif)
,. Suppose that
,
. There are two possible cases for
.If
for all
, similar to (Case 1), we can prove there must be a
such that
. Here we omit it. Let
. Then
for
and
. For
, we have
. So,
and
for
. If there exists a
such that
. Let
. Then
for
and
. For
, we have
.
Thus in both case the similar argument can be continued since for some
. This completes the proof.
Remark 3.7.
4. Numerical Analysis on Impulsive Perturbations
It is well known that the continuous system (1.3) cannot be solved explicitly. Thus we should study the system (1.3) by using numerical method and research the long-term behavior of the solutions to get more information about the dynamic behaviors of the system (1.3). We thus numerically investigate the influence of impulsive perturbation. For this, we fix the parameters except the control parameters ,
and
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ35_HTML.gif)
It is from [5] that the system (1.3) with and
has an unique limit cycle. Moreover, Figure 1 shows that the phase portrait of the system (1.3) with
,
, and
has a limit cycle too. From Theorem 3.2, we know that the prey-free periodic solution
is locally asymptotically stable provided that
. A typical prey-free periodic solution
of the system (1.3) is shown in Figure 2, where we observe how the variable
oscillates in a stable cycle while the prey
rapidly decreases to zero. On the other hand, if the amount
of releasing species is smaller than
, then the prey and the predator can coexist on a stable positive periodic solution (see Figure 1) and the system (1.3) can be permanent, which follows from Theorem 3.6.
Now we investigate the effect of impulsive perturbations. In Figure 3, we displayed bifurcation diagrams for the prey and predator populations as increasing from 0 to 13 with an initial value
. The resulting bifurcation diagram clearly show that the system (1.3) has rich dynamics including cycles, periodic doubling bifurcation, chaotic bands, periodic window, and period-halving bifurcation. Figures 4 and 5 are the magnified parts of Figure 3, and the windows of periodic behaviors are more visible.
As is evident from Figure 3, the solutions of the system (1.3) are -periodic when
and
and
-periodic when
. Generally, periodic doubling leads to chaos. We can take a local view of this phenomenon in Figure 4. But Figures 4 and 6 show the route to chaos through the cascade of period four. This phenomenon is caused by sudden changes when
. We can also find such phenomena when
, and so on. One of interesting things is that they can lead to non-unique attractors. In fact, Figure 7 exhibits the existence of multiattractors when
. These results show that just one parameter could give rise to multiple attractors. Narrow periodic windows and wide periodic windows are intermittently scattered (see Figure 3). At the end of the chaotic region, there is a cascade of period-halving bifurcation from chaos to one cycle. (see Figures 5 and 8). Periodic halving is the flip bifurcation in the opposite direction.
5. Discussion
In this paper, we have studied the effects of control strategies on a predator-prey system with Monod-Haldane type functional response. Conditions for the system to be extinct are given by using the Floquet theory of impulsive differential equation and small amplitude perturbation skills. Also, it is proved that the system the system (1.3) is permanent via the comparison theorem. Moreover, numerical examples on impulsive perturbations have been illustrated to substantiate our mathematical results and to show that the system we have considered in this paper gives birth to various kinds of dynamical behaviors.
Actually, in the real world, there are a number of environmental factors we should consider to describe the world more realistically. Among them, seasonal effect on the prey is one of the most important factors in the ecological systems. There are many ways to apply such phenomena in an ecological system [22, 23]. In this context, we think about the intrinsic growth rate in the system (1.3) as periodically varying function of time due to seasonal variation, which is superimposed as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ36_HTML.gif)
where the parameter represents the degree of seasonality,
is the magnitude of the perturbation in
and
is the angular frequency of the fluctuation caused by seasonality. Now, the system (1.3) can be changed as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ37_HTML.gif)
And then we get the following results via similar methods used in the previous sections.
Theorem 5.1.
Let be any solution of the system (5.2). Then the prey-free periodic solution
is locally asymptotically stable if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ38_HTML.gif)
Proposition 5.2.
There is an such that
for all
large enough, where
is a solution of the system (5.2).
Theorem 5.3.
The system (5.2) is permanent if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F598495/MediaObjects/13660_2010_Article_2201_Equ39_HTML.gif)
Thus, the seasonal effect on the prey may have also deeply influences on dynamics of the system (1.3).
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Acknowledgments
The first author is supported by Catholic University of Daegu Research Grant. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0004426).
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Kim, H., Baek, H. Dynamics of a Predator-Prey System Concerning Biological and Chemical Controls. J Inequal Appl 2010, 598495 (2010). https://doi.org/10.1155/2010/598495
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DOI: https://doi.org/10.1155/2010/598495