1. Introduction
To formulate the population dynamics, in 1963, MacArthur and Rosenzweig proposed the following predator-prey model with Michaelis-Menten functional response [1]
(1.1)
where the function
is the population of the predator at time t, the function
is the population of the prey at time t, m is the maximum growth (birth) rate of the predator, d is the death rate of the predator,
is the yield factor of the predator feeding on the prey and a is the half-saturation constant of the predator, which is the prey density at which the functional response of the predator is half maximal. The parameters
and K are the intrinsic rate of increase and the carrying capacity for the prey population, respectively. The parameters
and d are positive constants. Additionally, it is assumed that
for biological meanings.
Using the following rescaling
(1.2)
it follows that system (1.1) becomes
(1.3)
In this paper, it is assumed that
is sufficiently large; then
is a small parameter; therefore Equation (1.3) is a standard singularly perturbed system. From biological meanings, this assumption implies the prey population in the model (1.3) grows much faster than the predator population.
By switching to the fast time scale
(1.4)
one obtains the following equivalent system
(1.5)
If
, let
, then
and system (1.5) are rewritten as the follows
(1.6)
Accordingly, the results in [2] [3] [4] are reformulated in the following form:
1) If
, then the equilibrium
of system (1.5) is asymptotically stable.
2) If
, then the equilibrium
of system (1.6) is asymptotically stable.
3) If
, then the equilibrium
of system (1.6) is asymptotically stable.
4) If
, then the equilibrium
of system (1.6) is unstable and system (1.6) possesses a unique large-amplitude periodic solution.
In this paper, it is always assumed that
and
.
Additionally, it is also assumed
in system (1.6) for biological meanings.
By using the geometric singular perturbation theory, it will be proved in this paper that the canard explosion phenomenon happens in system (1.6) as the parameter
decreases through
. This canard explosion phenomenon can
explain the reason why the sudden transition from a small-amplitude periodic
solution, which bifurcates from the equilibrium
via the supercritical Hopf bifurcation at
, to a large-amplitude relaxation oscillation which emerges at
.
There are a great deal of articles [5] - [10], which are related to study the dynamics of predator-prey systems, such as bifurcations, stability, and so on.
Canard solutions were first analyzed by Benoit, Callot, Diener and Diener [11] using non-standard analysis in van der Pol equations. A canard solution is a solution of a singularly perturbed system which follows an attracting slow manifold, passes close to a non-hyperbolic point of the critical manifold, and then follows, rather surprisingly, a repelling slow manifold for a considerable amount of time before being repelled. The existence of a canard solution can lead to canard explosion, that is, a transition from a small limit cycle to a relaxation oscillation through a sequence of canard cycles upon variation of a parameter. Afterward, Eckhaus [12] studied the existence of canard solutions for van der Pol equation by employing the method of matched asymptotic expansion. A breakthrough in geometric explanation of canard cycles and canard explosion came with the work of Dumortier and Roussarie [13], who analyzed these phenomena in van der Pol’s equation by means of blow up technique and foliation of center manifolds in detail. From the work of Dumortier and Roussarie, it became apparent that blow up technique was the right tool for analyzing non-hyperbolic points of the slow manifold in a singularly perturbed system. Motivated by their work, Krupa and Szmolyan extended the standard normally hyperbolic geometric singular perturbation [14] [15] to non-hyperbolic points [16] [17] [18] by employing the blow up technique. Recently, canard solutions of a singularly perturbed system are extensively studied [19] - [24]. An introduction to basic knowledge on the geometric singular perturbation theory can be also founded in [25].
The paper is organized as follows. Section 2, Section 3 and Section 5 identify the fold point, the transcritical point, the Pitchfork Point and the canard point of system (1.6) respectively; Section 4 discusses Hopf bifurcations and relaxation oscillations of system (1.6); Section 6 analyzes the canard explosion phenomenon of system (1.6). Finally, some concluding remarks are given in Section 7.
2. Fold Point
Let
, system (1.6) becomes
(2.1)
Let
, then system (2.1) reduces to
(2.2)
where
and
.
Let
then system (2.2) can be rewritten as
(2.3)
Setting
in system (2.3) results in the layer problem
(2.4)
In term of the time rescaling
, system (2.3) becomes
(2.5)
Setting
in system (2.5) results in the reduced problem
(2.6)
Let
be the slow manifold, which consists of two parts
and
, where
and
Let
and
.
Let
and
Assume that
, then it can verified that
Therefore, by the definition of a fold point [16], (0, 0) is a fold point of system (2.3).
Under the assumption that
, it can be verified that the branch
is attracting and the branch
is repelling for the layer problem. The origin (0, 0) is nonhyperbolic, weakly attracting from the left and weakly repelling to the right. Moreover, the reduced flow on
and
is directed towards the fold point (0, 0), see Figure 1 for the dynamics of the layer problem and the reduced problem.
Figure 1. Slow-fast dynamics of system (2.3) for
in the case that
.
The standard normally hyperbolic geometric singular perturbation [14] implies that outside an arbitrarily small neighborhood of (0, 0), the manifolds
and
perturb smoothly to locally invariant manifolds
and
, which are simply solutions to system (2.3).
Let
be a section transverse to the fast fiber, where
is a suitable interval and
is a suitable constant, see Figure 1.
By theorem 2.1 in [16], it follows that
Proposition 2.1. There exists
such that for
, the manifold
passes through
at a point
with
.
3. Transcritical Point and Pitchfork Point
Let
, then system (2.2) becomes
(3.1)
Let
, then system (3.1) becomes
(3.2)
Let
then it follows that system (3.2) can be rewritten as
(3.3)
Under the assumption that
, it can be calculated that
It follows that
By the definition of a transcritical point in [17], it can be seen that the point (0, 0) is a transcritical point of system (3.3), which implies that the point
is a transcritical point of system (2.2).
Remark 3.1. If
, then
. Furthermore, it is can be calculated that
. By the definition of a pitchfork
point in [17], it can be seen that the point (0, 0) is a pitchfork point of system
(3.3), which implies that the point
is a pitchfork point of system (2.2).
If
, then it can be seen that the point
is also a transcritical point of system (2.2).
However,in the cases that
and
,canard explosion phenomena do not happen in (2.2).
Under the assumption that
, it can be verified that the branch
is attracting and the branch
is repelling for the layer problem. The point (1, 1) of system (2.2) is nonhyperbolic, weakly repelling from the left and weakly attracting to the right, see Figure 1.
The standard normally hyperbolic geometric singular perturbation [14] implies that outside an arbitrarily small neighborhood of (1, 1) of system (2.2), the manifolds
and
perturb smoothly to locally invariant manifolds
and
. In the following, it will be analyzed that how does
pass through
a neighbourhood of the transcritical point
of system (2.2).
Let
, then system (3.2) becomes
(3.4)
where
,
.
Letting
(3.5)
and substituting (3.5) into Equation (3.4), then by directly calculating and dropping the tildes, Equation (3.4) becomes
(3.6)
with
,
and
.
Therefore, by using a result in [17], it follows that
Proposition 3.2. There exists
and a function
with
such that for
, the slow manifold
extend to
for sufficiently small
.
4. Relaxation Oscillation and Hopf Bifurcation
Based on the local dynamics nearby the fold point (0, 0) and the transcritical
point
of system (2.2), the following result can be obtained.
Theorem 4.1. Assume that
, then for sufficiently small
, system (2.2) has a stable large-amplitude limit cycle
.
Proof. Let
be a section of the flow defined as a small horizontal interval
intersecting
at a point between
and (0,
0), see Figure 1. Consider tracking a trajectory starting in
for
. Initially this trajectory will be attracted to
and then pass beyond the fold point (0, 0) until it reach the section
. As this trajectory arrives in the vicinity of
, it will be attracted to
and then pass beyond the transcritical point (1, 1). Rather surprising, this trajectory will follow
for a considerable amount of time until it is repelled by
. Therefore this trajectory will come close to
and it will follow
until it reaches
. Let
be the return map. By the geometric singular perturbation theory, it follow that for
,
is a contraction map. By the implicit function theorem, there exists a unique and attracting fixed point of
in
. This fixed point gives rise to a stable large-amplitude limit cycle
.
Theorem 4.2. There exist
such that system (2.2) has a unique and stable small-amplitude limit cycle bifurcating from the equilibrium (0, 0) via the supercritical Hopf bifurcation for
.
Proof. System (2.2) has an equilibrium
. The linearization of system (2.2) at E has the following form
which has eigenvalues
If
, then
,
,
.
Lengthy calculations show that the first Liapunov coefficient
Therefore, the results in theorem 4.2 are justified.
5. Canard Point
In this section, it is assumed that
.
Let
then system (2.2) can be rewritten as the following form.
(5.1)
It can verified that
By the definition of a canard point in [16], it can be seen that the point
is a canard point of system (5.1).
The reduced dynamics on
is governed by the equation
(5.2)
It follows that the right-hand side of system (5.2) is a smooth function at the origin. Let
denote a maximal solution of system (5.2) with the property
. It follows that
exists and passes through the origin, see Figure 2 for the dynamics of the layer problem and the reduced problem.
By theorem 3.1 in an article [16], it follows that
Proposition 5.1. There exists
and a smooth function
defined on
such that for
, a solution starting in
connects to
if and only if
with
.
6. Canard Explosion
Let
Theorem 6.1. For any
, there exists a unique
such that
Figure 2. Slow-fast dynamics of system (2.2) for
in the case that
.
where
denotes the solution of system (2.5) on
at
.
Proof. For the limiting slow dynamics on
, system (2.5) is reduced to
Therefore, for any
, it can be calculated that
It follows that
which is equivalent to
Let
Then for any
, it follows that
Therefore, there exists a unique
such that
It follows that there exists a unique
such that
. Thus theorem 6.1 is proved.
Define a map
by
. where
denotes the solution on
at
and
is determined by theorem 6.1.
Remark 6.2. At
, define
.
Define singular canard cycles
for
,
for
. See Figure 3 for an illustration.
Remark 6.3. As
, the large-amplitude limit cycle
in theorem 4.1 converges to
at
in the Hausdorff distance.
Figure 3. Singular canard cycles
. Left:
for
. Right:
for
.
Figure 4. Blow-up singular cycles for
.
By blowing up the transcritical point
of
system (2.2) and the canard point (0, 0) of system (2.2), see Figure 4. The following results can be obtained by theorem 3.3 in [18].
Theorem 6.4. Fix
sufficiently small. Then for
, system (2.2) possesses a family of periodic orbits
whichis smooth in
,and such that:
1) As
,the family
converges uniformly in Hausdorff distance to
.
2) Any periodic orbit passing sufficiently close to the slow manifold is a member of the family
or a relaxation oscillation.
3) All canard cycles are stable and the function
ismonotonic in s.
7. Conclusions
As shown in this paper, canard explosion phenomenon in the predator-prey model with Michaelis-Menten functional response happens due to the interactions between the local dynamics nearby turning points, such as, fold point, transcritical point, canard point, and the global return mechanism induced by the slow manifold in system (1.6). Additionally, canard explosion phenomenon in two-dimensional singularly perturbed autonomous dynamical system is a codimension one bifurcating phenomenon, in which the parameter
is chosen as
a bifurcating parameter, as the parameter
decreases through
, the sudden transition from a small-amplitude periodic solution, which bifurcates from the equilibrium
via the supercritical Hopf bifurcation at
, to a large-amplitude relaxation oscillation which emerges at
, takes place by canard explosions.
However, the global return mechanism in the predator-prey model is slightly different from that in van der Pol’ equations analyzed by Krupa and Szmolyan [18]; the latter is S shape; the former is not S shape.
Additionally, canard explosion phenomenon in two dimensional singularly perturbed autonomous dynamical system is a codimension one bifurcating phenomenon. In this paper, the parameter
is selected as a bifurcating parameter, and canard explosion phenomenon in system (1.6) is demonstrated. Actually, the parameter
in system (1.5) can be also chosen as a bifurcating parameter, and it can be shown that canard explosion phenomenon happens in
system (1.5) as the parameter
decreases through
.
Acknowledgements
This work was supported by the NNSFC 11971477.