Impulsive Predator-Prey Dynamic Systems with Beddington-DeAngelis Type Functional Response on the Unification of Discrete and Continuous Systems ()
1. Introduction
The relationships between species and the outer environment, and the connections between different species are the description of the predator-prey dynamic systems which is the subject of mathematical ecology in biomathematics. Various types of functional responses in predator-prey dynamic system such as Monod-type, semi-ratio- dependent and Holling-type have been studied. [1] is an example for the study about Holling-type functional response. In this paper, we consider the predator-prey system with Beddington DeAngelis type functional response and impulses. This type of functional response first appeared in [2] and [3] . At low densities this type of functional response can avoid some of the singular behavior of ratio-dependent models. Also predator feeding can be described much better over a range of predator-prey abundances by using this functional response.
In a periodic environment, significant problem in population growth model is the global existence and stability of a positive periodic solution. This plays a similar role as a globally stable equilibrium in an autonomous model. Therefore, it is important to consider under which conditions the resulting periodic nonautonomous system would have a positive periodic solution that is globally asymptotically stable. For nonautonomous case there are many studies about the existence of periodic solutions of predator-prey systems in continuous and discrete models based on the coincidence theory such as [4] -[12] .
Impulsive dynamic systems are also important in this study and we try to give some information about this area. Impulsive differential equations are used for describing systems with short-term perturbations. Its theory is explained in [13] -[15] for continuous case and also for discerete case there are some studies such as [16] . Impulsive differential equations are widely used in many different areas such as physics, ecology, and pest control. Most of them use impulses at fixed time such as [17] [18] . By using constant functions, some properties of the solution of predator-prey system with Beddington-DeAnglis type functional response and impulse impact are studied in [19] for continuous case.
In this study unification of continuous and discrete analysis is also significant. To unify the study of differential and difference equations, the theory of Time Scales Calculus is initiated by Stephan Hilger. In [20] [21] , unification of the existence of periodic solutions of population models modelled by ordinary differential equations and their discrete analogues in form of difference equations, and extension of these results to more general time scales are studied.
The unification of continuous and discrete case is a good example for the modeling of the life cycle of insects. Most of the insects have a continuous life cycle during the warm months of the year and die out in the cold months of the year, and in that period their eggs are incubating or dormant. These incubating eggs become new individuals of the new warm season. Since insects have such a continuous and discrete life cycle, we can see the importance of models obtained by the time scales calculus for the species that have unusual life cycle. Therefore, in this paper we try to generalize periodic solutions of predator-prey dynamic systems with Beddington-DeAn- glis type functional response and impulse to general time scales.
2. Preliminaries
Below informations are from [20] . Let X, Z be normed vector spaces,
be a linear mapping,
be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if
and ImL is closed in Z. If L is a Fredholm mapping of index zero and there exist continuous projections
and
such that
,
, then it follows that
is invertible. We denote the inverse of that map by
. If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω if
is bounded and
is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism
.
The above informations are important for the Continuation Theorem that we give below.
Theorem 1. (Continuation Theorem). Let L be a Fredholm mapping of index zero and N be L-compact on Ω. Suppose
(a) For each
, every solution z of
is such that
;
(b)
for each
and the Brouwer degree
Then the operator equation
has at least one solution lying in
.
We will also give the following lemma, which is essential for this paper.
Lemma 1. Let
and
. If
is w-periodic, then
![]()
3. Main Result
The equation that we investigate is:
(1)
,
,
,
,
,
,
, ![]()
and
Here
is periodic, i.e
if
then
and
, ![]()
![]()
,
, , and
Each functions are from ![]()
Lemma 2. If
and
then all positive solutions
are tends to 0 as t tends to infinity.
Proof. If we using the first equation of (1) we obtain,
![]()
Since
Hence ![]()
Similarly ![]()
Theorem 2. In addition to conditions on coefficient functions
If
![]()
and
![]()
then there exist at least a w-periodic solution.
Proof.
with the norm:
![]()
and
![]()
with the norm:
![]()
Let us define the mappings
and
by
such that
![]()
and
such that
![]()
Then
,
and
are constants.
![]()
is closed in
and
, therefore
is a Fredholm mapping of index zero.
There exist continuous projectors
and
such that
![]()
and
![]()
where ![]()
The generalized inverse
is given,
![]()
![]()
Let
![]()
![]()
![]()
and
![]()
![]()
Clearly,
and
are continuous. Since
and
are Banach spaces, then by using Arzela-
Ascoli theorem we can find
is compact for any open bounded set
Addition-
ally,
is bounded. Thus,
is L-compact on
with any open bounded set ![]()
To apply the continuation theorem we investigate the below operator equation.
(2)
Let
be any solution of system (2). Integrating both sides of system (2) over the interval
we obtain,
(3)
From (2) and (3) we get
(4)
where ![]()
(5)
where ![]()
Note that since
and there are q impulses which are constant, then there exist
,
such that
(6)
(7)
By the second equation of (3) and (6) and the first assumption of Theorem 2, we have
![]()
and
where ![]()
Using the second inequality in Lemma 1 we have
(8)
By the first equation of (3) and (6) we get
where
![]()
using the first inequality in Lemma 1 and (4), we have
(9)
By (8) and (9)
Using (9), second equation of (3) and first equation of (7), we can derive that
![]()
Therefore
![]()
By the assumption of the theorem we can show that
and
where ![]()
Hence, by using the first inequality in Lemma 1 and the second equation of (3),
(10)
We can also derive from the second equation of (3) that
![]()
![]()
Again using second assumption of Theorem 2 we obtain
![]()
and
where ![]()
By using the second inequality in Lemma 1 and (5), we obtain
(11)
By (10) and (11) we have
Obviously,
and
are both inde-
pendent of
. Let
. Then
Let
and
verifies the requirement (a) in Theorem 1. When
,
is a constant with
then
![]()
![]()
where
such that ![]()
Define the homotopy
where
![]()
Take
as the determinant of the jacobian of G. Since
, then jacobian of G is
![]()
All the functions in jacobian of G is positive then
is always positive. Hence
![]()
Thus all the conditions of Theorem 1 are satisfied. Therefore system (1) has at least a positive w-periodic solution.
Theorem 3. If same conditions are valid for the coefficient functions in system (1) and
![]()
is satisfied then there exist at least a w-periodic solution.
Proof. First part of the proof is very similar with the proof of Theorem 2. By (2), (3) and (6)
![]()
By (3)
Also by the assumption of Theorem 3
Then we get
.
And using the second inequality in Lemma 1 we have
(12)
By the first equation of (3) and (6)
![]()
Then we get
where ![]()
Using the first inequality in Lemma 1 we have
(13)
By (12) and (13)
From the second equation of (3) and the second equation of (7), we can derive that
![]()
Therefore
![]()
Since
then
where ![]()
Hence, by using the first inequality in Lemma 1 and the second equation of (3),
(14)
By the assumption of Theorem 3 there exists
such that ![]()
![]()
is true. We need to get
such that
Let us assume there exists
such
that
Then by using (6) and (7) we obtain
![]()
If such t, s does not exists then
. Also from the first equation of (3), we have
![]()
By using first inequality in Lemma 1, we have
, where
![]()
Using the second equality in (3) and the assumption of the Theorem 4, we obtain
![]()
This implies
where
![]()
Hence, according to the above discussion we have
Using second inequality
in Lemma 1 we have
where ![]()
Thus
Obviously,
and
are both independent of
. Let
. Then
Let
then Ω verifies the requirement
(a) in Theorem 1. Rest of the proof is similar to Theorem 2.
Let there are two insect populations (one of them the predator, the other one the prey) both continuous while in season (say during the six warm months of the year), die out in (say) winter, while their eggs are incubating or dormant, and then both hatch in a new season, both of them giving rise to nonoverlapping populations. This situation can be modelled using the time scale
![]()
Here impulsive effect of the pest population density is after its partial destruction by catching, poisoning with chemicals used in agriculture (can be shown by
) and impulsive increase of the predator population density is by artificially breeding the species or releasing some species
. In addition to these, if the model assumes a BeddingtonDeAngelis functional response as in (1) and if the assumptions in Theorem 2 or 3 are satisfied then there exists a 1-periodic solution of (1).
Corollary 1. If
in the system (1) and
![]()
is satisfied then the system (1) has at least one w-periodic solution.
Example 1.
k start with 0.
![]()
Impulse points:
,
and
.
,
,
Example 1 satisfies all the conditions of Theorem 2, thus it has at least one periodic solution.
Example 2.
k start with 0.
![]()
Impulse points:
,
and
.
,
,
Example 2 satisfies all the conditions of Theorem 3, thus it has at least one periodic solution.
Theorem 4. If all the coefficient functions in system (1) is positive, w-periodic, from
and impulses are 0; also
![]()
is satisfied then there exist at least a w-periodic solution. ![]()
Proof. First part of the proof is similar to Theorem 2, only difference is the zero impulses. If the assumption of Theorem 4 is true then there exists
such that for all ![]()
![]()
is satisfied. Suppose there exist
such that
. Then similar to proof of Theorem 4 we can find
.
If such s, t does not exist
. Using the first equation of (1) and assuming
is the minimum of
. Then
![]()
Thus we get
![]()
Then ![]()
If
is a right dense point then
If
is right scattered, we interested
with the maximum of the solution. Let
be the maximum of x(t).
![]()
Then
If
, then ![]()
If
, then ![]()
Thus
![]()
Using (3) and (7) above results we obtain
![]()
This implies
![]()
Hence, according to the above discussion we have
Using second inequality in
Lemma 1 we have
Thus
Rest of the proof
is similar to Theorem 2.
Corollary 2. In Theorem 4 if we take
as
then we get Theorem 3 in [21] .
Example 3.
k start with 0.
![]()
Example 3 satisfies all the conditions of Theorem 4, thus it has at least one periodic solution.
All the graphs that we see in Figures 1-3 are obtained by Mathlab.
4. Discussion
In this paper, the impulsive predator-prey dynamic systems on time scales calculus are studied. We investigate when the system has periodic solution. Furthermore, three different conditions have been found which are necessary for the periodic solution of the predator-prey dynamic systems with Beddington-DeAngelis type functional response. Also by using graphs, we are able to show that the conditions that are found in Theorem 2, 3
![]()
Figure 1. Numeric solution of Example 1 shows the periodicity.
![]()
Figure 2. Numeric solution of Example 2 shows the periodicity.
![]()
Figure 3. Numeric solution of Example 3 shows the periodicity.
and 4 are enough for the periodic solution of the given system. In this work, since our system can model the life cycle of the such species like insects, what we have done new is finding necessary condition for the periodic solution of the given predator-prey system with sudden changes. In addition to these, according to the structure of the given time scale
, the conditions that are found in Theorem 2, 3 and 4 become useful.