Dynamic Analysis of a Predator-Prey Model with Holling-II Functional Response ()
1. Introduction
Predator-prey system refers to the relationship in which one species is preyed upon by another species in a community of organisms. This relationship is an important part of the biome and can affect the number, distribution and composition of species, thereby altering the stability and integrity of the environment.
The Holling-II model (also known as the Lotka-Volterra model or food chain model) is a classic dynamical model used to study predator-prey interactions. The model is based on the assumption that as the number of prey increases, so does the number of predators; however, as an increase in predators, the likelihood of prey being preyed on increases, leading to a decrease in the number of prey. In turn, the decrease in prey populations inhibits the growth of predator populations. From the assumption Holling-II model, it can be seen that the quantitative relationship between predator and prey is nonlinear. In recent years, many researchers have studied predator-prey dynamics model from different angles (see [1] - [11] ). In the literature [10] , the authors studied a Holling-II predatory functional response in a predator-prey model as follows
(1)
They investigated the stabilities of the equilibrium point and the existence of limit cycle. In literature [11] , the authors considered a Leslie-Gower predator-prey model with time delay and linear harvesting term
(2)
where
and
denote the linear capture terms caused by external interference, and
and
. They studied the positive steady-state properties of the model (2) and gave the conditions for Hopf bifurcation near the positive equilibrium point. Mainly inspired by literatures [10] and [11] , combining models (1) and (2), we propose the following predator-prey model with Holling-II type functional response and a linear capture term
(3)
where u and v represent the densities of prey and predator populations at time t respectively, a, b, d1 and d2 are positive parameters, and h1 and h2 are negative parameters. We use differential equation theory to discuss the existence and the stabilities of non-negative equilibrium points of the model (3). The results show that under certain limited conditions, these two populations can maintain balance.
2. The Equilibria of the System Model
In view of the practical significance of the system, we only consider the case where u and v satisfy
. Define
Let
(4)
Then, we have three equilibrium points:
,
,
where
satisfy
We easily know that
and
when
.
3. Stabilities of the Equilibrium Points
The linear approximation at any point
of the model (3) is as follows
(5)
where
,
,
,
.
3.1. Stability of Equilibrium Point
In this case, we have
,
,
,
.
So the linear approximation Equation (5) is as follows:
(6)
The coefficient matrix of Equation (5) is
. Then the characteristicroots are
and
. Because
, the following conclusion is obvious.
Theorem 1. If
, then
is an unstable saddle point. If
, then
is an asymptotically stable node.
3.2. Stability of Equilibrium Point
In this case, we have
,
,
,
.
So the linear approximation Equation (5) is:
(7)
The coefficient matrix of Equation (7) is
The characteristic roots are
,
.
Therefore, the following conclusion is obtained:
Theorem 2. If
, then
, so
is an unstable saddle point. If
, then
, so
is an asymptotically stable node.
3.3. Stability of Plane Equilibrium Point
From
,
we have that
,
,
,
.
Therefore, the coefficient matrix of Equation (5) for
is
and its characteristic equation is
Let
,
, (8)
then we have
. (9)
Let
,
. (10)
Then the characteristic roots of Equation (9) satisfy
,
.
From (8) and (10), we have
Define
,
then, its discriminant is
.
Thus, equation
has two roots:
.
Obviously, we have
,
.
Therefore, we can get
(i) as
, we have
, which implies
;
(ii) as
, we have
, which implies
;
(iii) as
, we have
, which implies
.
From (8) and (10), we also have
Define
then, its discriminant is
From condition of
, we can know that
,
.
Therefore, the following relationships between q and
hold,
(a) If
, then
for all
, and thus
;
(b) If
, then
has two equal roots:
,
and
.
Therefore,
for either
or
, and thus
;
(c) If
, Then
has two real roots:
and
,
.
Therefore, as
, we have
, which implies
; as
or
, we have
, which implies
.
We give some assumptions as the follows.
(H1)
;
(H2)
;
(H3)
.
Theorem 3. Suppose conditions (H1) and
hold, then
is an unstable saddle point.
Proof If condition (H1) holds, we know
. Combined with condition
and above discussion, we get
. It implies that Equation (9) has two real roots which have different signs, thus
is an unstable saddle point.
Theorem 4. Suppose conditions (H1) holds, we have
(A) if
or
, then
is an unstable node or unstable focus;
(B) if
or
, then
is a stable node or stable focus;
(C) if
, then
is a center.
Theorem 5. Suppose conditions (H2) holds, we have
(A) if
and
, then
is an unstable node or unstable focus;
(B) if
and
, then
is a stable node or stable focus;
(C) if
, then
is a center.
Theorem 6. Suppose conditions (H3) holds, we have
(A) if
, then
is an unstable node or unstable focus;
(B) if
, then
is a stable node or stable focus;
(C) if
, then
is a center.
4. Conclusion
This paper investigates a Holling-II functional response predator-prey model with linear capture terms. The existence of non-negative equilibrium point of the model was discussed, and the stability of the equilibrium point was analyzed. It is clear from this paper that human capture intensity can maintain an equilibrium position for both populations under certain constraints. However, when the intensity of human capture exceeds a certain limit, the two populations will lose equilibrium, which will have a negative impact on species diversity and environmental stability. In other words, human capture behavior, within certain limits, can not only maintain human needs but also achieve sustainable development of ecological resources. This paper provides a theoretical reference for the decision-making of relevant departments.
Acknowledgements
This work has been supported by the NSF of China (Grant No. 11961021), the NSF of Guangdong Province (Grant No. 2022A1515010964, 2022A1515010193), the Innovation and Developing School Project (Grant No. 2019KZDXM032) and the Key Project of Science and Technology Innovation of Guangdong College Students (Grant No. pdjh2022b0320, pdjh2023b0325).