西 南 交 通 大 学 学 报
第 54 卷第 6 期
2019 年 12 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 54 No. 6
Dec. 2019
ISSN: 0258-2724 DOI:10.35741/issn.0258-2724.54.6.5
Research Article Mathematics
T
HE
D
YNAMICAL
B
EHAVIOR OF
S
TAGE
S
TRUCTURED
P
REY
-P
REDATOR
M
ODEL IN THE
H
ARVESTING AND
T
OXIN
P
RESENCE
分阶段结构的捕食与被捕食模型的动力学行为
Azhar Abbas Majeed *, Moayad H. Ismaeel
Department of Mathematics, College of Science, University of Baghdad
Al-Jadriya, Karrada, Baghdad, 10071, Iraq, azhar_abbas_m@scbaghdad.ed, moaed1983@yahoo.com
Abstract
In this study, a mathematical model that consists of a form of prey-predator system with stage structure in the presence of harvesting and toxicity has been proposed and studied by using the classic Lotka-Volterra functional response. The presence, uniqueness, and boundedness resolution of the suggested model are discussed. The steadiness enquiries of all possible stability points tare studied. The global steadiness of these stability points are accomplished by fitting Lyapunov functions. As a final point, numerical models are put through not just for conforming tthe hypothetical results attained, but also to demonstrate the influences of distinction of each factor on the suggested paradigm.
Keywords: Functional Response, Prey-Predator, Stability Analysis, Stage-Structure, Lyapunove Function.
摘要 在这项研究中,通过使用经典的洛特卡 - 沃尔泰拉功能响应,提出并研究了一个数学模型, 该模型由具有阶段结构的捕食者-捕食者系统组成,并且存在收获和毒性。 讨论了所建议模型的 存在性,唯一性和有界分辨率。 研究了所有可能的稳定性点的稳定性查询。 这些稳定性点的全局 稳定性是通过拟合利亚普诺夫函数来实现的。最后,不仅要通过数值模型来验证所获得的假设结 果,而且还要证明每个因素的不同对建议范式的影响。 关键词: 功能响应,捕食者,稳定性分析,阶段结构,李雅普诺夫函数。
I. I
NTRODUCTIONThe prey-predator system is one of the most important topics in the ecosystem. It is used to solve many complex problems which cannot be predicted. Thus, it is considered an alternative method for improving our knowledge of the physical and biological processes related to the
environment. One of the most serious problems that threaten the ecosystem is over-harvesting of living things, because of the massive population increase and the desire for people to obtain more resources that led to specific dangers to the ecosystem and has become a problem. Several models were proposed according to harvest
models [1], [2]. While many researchers have tried to limit this problem by suggesting a model containing a refuge to save the prey from extinction due to over-harvesting, and predation for8example [3], [4]. On the other hand, the age factor has had a significant impact on the rate of growth and reproduction. In recent years, many prey-predator models based on age-structure have been studied by authors [5], [6], [7]. The other major problem affecting the ecosystem is pollution caused by toxic substances. Many studies have considered the environmental effects of toxic substances, Hallam and Clark [8] studied the effects of toxic substances on exposed populations. In addition, Hallam and De Luna [9] have discussed the effects of a toxin through the food chain of the population. While Friedman and Shukla [10] developed the Models of predator-prey systems in a polluted closed environment with single species. Chattopadhyay [11] studied the effects of toxic substances on two competing species and noted that the toxic substances have some stabilizing effects on the system. Montoya et al. [12] considered two types of factors, such as anti-predator behavior and group defense of stage-structure model. There is no doubt that the presence of toxicity will affect the harvest, as shown by some studies that focused on the existence of toxic substances in harvests [13], [14], [15], [16]. Finally, Majid [17] suggests a model that contains stage structures in both populations with the effect of toxicants. In this paper, the stage-structured prey-predator model with harvesting and toxicity has been proposed and studied. The considered model consists of four nonlinear common differential equations to define the interactions by using the Lotka-Volterraa type of functional response. This system is analyzed by using the linear stability to find the conditions for which the feasible equilibrium points are stable. Global stability conditions for the proposed model are described by using appropriate Lyapunove functions.
II. M
ODELF
ORMULATIONSIn this section, the model consists of two species of prey and predator, with each species divided into two classes: immature and mature, which are denoted to their populations sizes at time T by , 1, and
respectively. Here, in order to frame the subtleties of such a scheme, the subsequent presumptions are contemplated:
The immature classes of prey and predator grew up to be mature with growth rates of and respectively. The immature prey depends completely on its feeding on mature prey that
grows logistically with an intrinsic growth rate of and a carrying capacity of in the absence of a mature predator. Also the immature predator depends completely on its feeding on a mature predator that consumes the immature and mature prey with the classical Lotka-Volterra functional response with consumption rates of and respectively. Therefore, the predator species grows due to attacks by mature predators on immature and mature prey with conversion rates of and . However, in the absence of prey species, the predator species decay exponentially with the mortality rates of and of immature and mature predator respectively. Moreover, the immature predator cannot attack any of the prey species; rather, it depends completely on his parents, so that it feeds on the portion of the consumed food by the mature predator from the first and second prey species with portion rates of and respectively. Finally, =1, 2, 3, 4 are the catchability coefficients and the toxicity coefficients of prey species and predator species respectively. According to the above assumptions, the model is formulated as follows:
(1)
In order to simplify the system, the number of parameters is reduced by using the following dimensionless variables and parameters:
Then dimensional system (1) becomes:
(2)
Obviously, the interaction functions of the scheme are unremitting and have unremitting partial derivatives on the subsequent positive four dimensional space:
=,{
, and hence the existence and uniqueness of the solution for system . Further, all the solutions of system with non-negative initial conditions are uniformly bound, as shown in the following theorem:
Theorem 1: All the solutions of system (2) are
uniformly bound.
Proof. Let be any
solution of the system (2)
with Now consider a
function: ,
and then take the time derivative of function: along with the solution of the system (2). So, due to the fact that the conversion rate constant from immature and mature prey population to mature and immature predator population cannot exceed the maximum predation rate constant from mature predator population to immature and mature prey population, hence from the biological point of view, always:
and , we get:
Now by solving this differential inequality for the initial value we get:
Hence all the solutions of system (2) are uniformly bound.
III. T
HES
UBSISTENCE OFE
QUILIBRIUMP
OINTSIn this subdivision, the subsistence of all possible equilibrium points of scheme (2) is debated. It is observed that scheme (2) has at most three non-negative equilibrium points, which are in the following:
- The equilibrium point always exists.
- The8equilibrium8point
exists8uniquely8in if the following condition holds:
- Finally, the positive equilibrium point exists if the following condition holds:
IV. T
HES
TABILITYA
NALYSISIn this section, the local stability analysis of system around each of the above equilibrium
points is discussed through computing the Jacobean matrix of system (2):
- The characteristic polynomial of the Jacobean matrix of system at
gives the four eigen values of with negative real parts provided that the following condition holds:
Then is locally asymptotically stable in under the condition . However, it is a saddle point (unstable) otherwise.
- The characteristic polynomial of the Jacobian matrix of system at
gives the four eigen values of with negative real parts due to the following conditions:
Hence, is locally asymptotically stable in under the conditions (6-8). However, it is a saddle (unstable) point otherwise.
- Finally, then the characteristic equation of the Jacobean matrix of system at is given by:
with
Now by using Routh-Hawiritiz criterion equation, has roots (eigen values) with negative real parts if and only if i = 1, 3,
4 and Clearly,
Hence, will be positive if in addition of conditions ( - ). Therefore, all the eigen values of have negative real parts under the given conditions and hence is locally asymptotically stable. However, it is unstable otherwise.
V. G
LOBALS
TABILITYA
NALYSES In this section, the global stability analysis for the equilibrium points that are locally asymptotically stable of system (2) is studied analytically with the help of Lyapunov method. We get:- Assume8that is locally asymptotically stable in . Then is globally asymptotically stable on the region
where
- Assume that is locally asymptotically stable in . Then is globally asymptotically stable on the region
that satisfies the following conditions:
- Assume8that of8system is locally asymptotically stable in the . Then
is a globally asymptotically stable on any region that satisfies the following conditions:
VI. N
UMERICALA
NALYSIS OFS
YSTEMIn this section, the dynamic behavior of system (2) is studied numerically for one set of parameters and different sets of initial points. The objectives of this study are first to investigate the effect of varying the value of each parameter on the dynamic behavior of system (2), and second to confirm our obtained analytical results.
It is observed that, for the following set of hypothetical parameters that satisfies stability conditions of the positive equilibrium point, system (2) has a globally asymptotically stable positive equilibrium point, as shown in Figure 1.
Figure 1. The8time series of8the8solution of the system
started from8the three8different initial points
and
, for the data8given by
the8trajectories8of as8a8function8of time, the8trajectories8of as8a function8of time,
trajectories8of as a8function8of time, the8trajectories8of as8a8function8of8time
Figure 1 shows that system has a globally asymptotically stable solution of the system approaches asymptotically to the positive
equilibrium point
starting from three different initial points, and this confirms our obtained analytical results. Now, in order to discuss the effect of the parameter values of the system on the dynamical behavior of the system, the system is solved numerically for the data given by Eq. by varying one parameter at a time.
Further, by varying one parameter at a time, it is observed that varying the parameter values, ,
and , do
not have anyeffect on the dynamic behavior of system (2) and the solution of the system still approaches the positive equilibrium point Varying in the range
, causes extinction of all species and the solution of system approaches asymptotically to , as shown in Figure 2a, for typical value , while the increase of this parameter in the range
the solution of system (2) approaches asymptotically to in the int. of , as shown in Figure 2b, for typical value further increasing this parameter further in the range
the solution of system approaches asymptotically to the equilibrium point in the int. of , as shown in Figure 2c, for typical value
.
Figure 2. (a) T Time8series of8the solution8of system8
for8the data given by with which
approaches to , Timewseries of thee
solution of systemy for the dataagiven bya with
which approaches.to ,
(c): Time.series of theasolution of systemw for the
data.given bya witha whichs approachesa to
in the int. of
While diversifying the parameter and maintaining the remnant parameters as in the data given in in the range it was observed that the solution of system approaches asymptotically. However, increasing this parameter in the range causes the extinction of the predator species and the solution of system approaches asymptotically to in the int. of , which increases the range to and causes the extinction of all the species. Further, the solution of system
approaches asymptotically to
The effect of altering the parameter , with , and keeping the rest of the parameters as in the data given in is observed in the solution of system approaches asymptotically to , while the increase of this parameter to
leads the solution of system to approach asymptotically. Moreover, in keeping with the rest of the parameters’ values as in the data given in with , the solution of system approaches asymptotically while the increase of this parameter to leads the solution of system to approach asymptotically. Finally, the parameters , and have the same effect on the behavior of the solution of system (2), and in keeping with the rest of the parameters as in the data given in in the range it is observed that the solution of system still approaches asymptotically, while the increase of this parameter to leads the solution of system to approach asymptotically.
VII. C
ONCLUSIONS ANDD
ISCUSSION In this work, we proposed and analyzed an ecological model that describes the dynamic behavior of the stage-structured model of prey– predator in both species with harvesting and toxicity. The model included four non-linear autonomous differential equations that describe the dynamics of four different populations, namely, first immature prey , mature prey , immature predator and mature predator . The boundedness of system (2) was discussed, and the existence conditions of all possible equilibrium points were obtained. The local as well as global stability analyses of these points were carried out. Finally, numerical simulation was used to specify the control set of parameters that affect the dynamics of the system and confirm our obtained analytical results. Therefore, system (2) was solved numerically for different sets of initial points and a set of parameters that started with the hypothetical set of data given by eq. (21), and the following observations were obtained. The system within the set of parameters imposed does not have a periodic solution. For the set hypothetical parameter value given in (21), the system (2) approaches asymptotically to the globally stable positive point Further, while altering one parameter at a time, it wasobserved that altering the parameters’ values, ,
and did
not have any effect on the dynamic behavior of system (2) and the solution of the system still approached The parameters and have a bifurcation with the values
, respectively. Finally, the parameters , and have a bifurcation with the values 1.226, 0.106 and =
= = 0.251, respectively.