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Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion System on Time Scales
Advances in Difference Equations volume 2009, Article number: 141589 (2009)
Abstract
This paper investigates the existence of periodic solutions of a ratio-dependent predator-prey diffusion system with Michaelis-Menten functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain suffcient criteria for the existence of periodic solutions for the system. Moreover, when the time scale is chosen as
or
, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.
1. Introduction
The traditional predator-prey model has received great attention from both theoretical and mathematical biologists and has been studied extensively (e.g., see [1–4] and references therein). Based on growing biological and physiological evidences, some biologists have argued that in many situations, especially when predators have to search for food (and therefore, have to share or compete for food), the functional response in a prey-predator model should be ratio-dependent, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. Starting from this argument and the traditional prey-dependent-only mode, Arditi and Ginzburg [5] first proposed the following ratio-dependent predator-prey model:

which incorporates mutual interference by predators, where is a Michaelis-Menten type functional response function. Equation (1.1) has been studied by many authors and seen great progress (e.g., see [6–11]).
Xu and Chen [11] studied a delayed two-predator-one-prey model in two patches which is described by the following differential equations:

In view of periodicity of the actual environment, Huo and Li [12] investigated a more general delayed ratio-dependent predator-prey model with periodic coefficients of the form

In order to consider periodic variations of the environment and the density regulation of the predators though taking into account delay effect and diffusion between patches, more realistic and interesting models of population interactions should take into account comprehensively other than one or two aspects. On the other hand, in order to unify the study of differential and difference equations, people have done a lot of research about dynamic equations on time scales. The principle aim of this paper is to systematically unify the existence of periodic solutions for a delayed ratio-dependent predator-prey system with functional response and diffusion modeled by ordinary differential equations and their discrete analogues in form of difference equations and to extend these results to more general time scales. The approach is based on Gaines and Mawhin's continuation theorem of coincidence degree theory, which has been widely applied to deal with the existence of periodic solutions of differential equations and difference equations.
Therefore, it is interesting and important to study the following model on time scales :

with the initial conditions

where . In (1.4),
represents the prey population in the
th patch
, and
represents the predator population.
is the prey for
, and
is the prey for
so that they form a food chain.
denotes the dispersal rate of the prey in the
th patch
. For the sake of generality and convenience, we always make the following fundamental assumptions for system (1.4):
-
(H)
are all rd-continuous positive periodic functions with period
;
are nonnegative constants.
In (1.4), set . If
, then (1.4) reduces to the ratio-dependent predator-prey diffusive system of three species with time delays governed by the ordinary differential equations

If , then (1.4) is reformulated as

which is the discrete time ratio-dependent predator-prey diffusive system of three species with time delays and is also a discrete analogue of (1.6).
2. Preliminaries
A time scale is an arbitrary nonempty closed subset of the real numbers
. Throughout the paper, we assume the time scale
is unbounded above and below, such as
and
. The following definitions and lemmas can be found in [13].
Definition 2.1.
The forward jump operator , the backward jump operator
, and the graininess
are defined, respectively, by

If , then
is called right-dense (otherwise: right-scattered), and if
, then
is called left-dense (otherwise: left-scattered).
If has a left-scattered maximum
, then
; otherwise
. If
has a right-scattered minimum
, then
; otherwise
.
Definition 2.2.
Assume is a function and let
. Then one defines
to be the number (provided it exists) with the property that given any
, there is a neighborhood
of
such that

In this case, is called the delta (or Hilger) derivative of
at
. Moreover,
is said to be delta or Hilger differentiable on
if
exists for all
. A function
is called an antiderivative of
provided
for all
. Then one defines

Definition 2.3.
A function is said to be rd-continuous if it is continuous at right-dense points in
and its left-sided limits exists (finite) at left-dense points in
. The set of rd-continuous functions
will be denoted by
.
Definition 2.4.
If ,
, and
is rd-continuous on
, then one defines the improper integral by

provided this limit exists, and one says that the improper integral converges in this case.
Definition 2.5 (see [14]).
One says that a time scale is periodic if there exists
such that if
, then
. For
, the smallest positive
is called the period of the time scale.
Definition 2.6 (see [14]).
Let be a periodic time scale with period
. One says that the function
is periodic with period
if there exists a natural number
such that
,
for all
and
is the smallest number such that
.
If , one says that
is periodic with period
if
is the smallest positive number such that
for all
.
Lemma 2.7.
Every rd-continuous function has an antiderivative.
Lemma 2.8.
Every continuous function is rd-continuous.
Lemma 2.9.
If and
, then
-
(a)
;
-
(b)
if
for all
, then
;
-
(c)
if
on
, then
.
Lemma 2.10.
If , then
is nondecreasing.
Notation.
To facilitate the discussion below, we now introduce some notation to be used throughout this paper. Let be
-periodic, that is,
implies
,

where is an
-periodic function, that is,
for all
,
.
Notation.
Let be two Banach spaces, let
be a linear mapping, and let
be a continuous mapping. If
is a Fredholm mapping of index zero and there exist continuous projectors
and
such that
,
, then the restriction
is invertible. Denote the inverse of that map by
. If
is an open bounded subset of
, the mapping
will be called
-compact on
if
is bounded and
is compact. Since
is isomorphic to
, there exists an isomorphism
.
Lemma 2.11 (Continuation theorem [15]).
Let be two Banach spaces, and let
be a Fredholm mapping of index zero. Assume that
is
-compact on
with
is open bounded in
. Furthermore assume the following:
-
(a)
for each
;
-
(b)
for each
;
-
(c)
.
Then the operator equation has at least one solution in
.
Lemma 2.12 (see [16]).
Let . If
is
-periodic, then

3. Existence of Periodic Solutions
The fundamental theorem in this paper is stated as follows about the existence of an -periodic solution.
Theorem 3.1.
Suppose that (H) holds. Furthermore assume the following:
-
(i)
-
(ii)
,
-
(iii)
,
-
(iv)
,
then the system (1.4) has at least one -periodic solution.
Proof.
Consider vector equation

Define

where is the Euclidean norm. Then
and
are both Banach spaces with the above norm
. Let
. Then

and . Since
is closed in
, then
is a Fredholm mapping of index zero. It is easy to show that
are continuous projectors such that
. Furthermore, the generalized inverse (to
)
exists and is given by
, thus

Obviously, are continuous. Since
is a Banach space, using the Arzela-Ascoli theorem, it is easy to show that
is compact for any open bounded set
. Moreover,
is bounded, thus,
is
-compact on
for any open bounded set
. Corresponding to the operator equation
, we have

Suppose that is a solution of (3.5) for certain
. Integrating on both sides of (3.5) from
to
with respect to
, we have




It follows from (3.5) to (3.9) that




Multiplying (3.6) by and integrating over
gives

which yields

By using the inequality , we have

Then

By using the inequality , we derive from (3.17) that

Similarly, multiplying (3.7) by and integrating over
, then synthesize the above, we obtain

It follows from (3.18) and (3.19) that

so, there exists a positive constant such that

which together with (3.19), there also exists a positive constant such that

This, together with (3.11), (3.12), and (3.21), leads to

Since , there exist some points
, such that

It follows from (3.21) and (3.22) that

From (3.8) and (3.9), we obtain that

This, together with (3.12), (3.13), and (3.26), deduces

From (3.6) and (3.24), we have

From (3.7) and (3.24), it yields that

Noticing that , from (3.8) and (3.9), deduces

There exist two points such that

Hence,

where . Then, this, together with (3.12), (3.13), (3.23), (3.28), (3.29), and (3.32), deduces

It follows from (3.27) to (3.33) that

From (3.34), we clearly know that are independent of
, and from the representation of
, it is easy to know that there exist points
such that
, where

Take , where
is taken sufficiently large such that
, and such that each solution
of the system
satisfies
if the system (3.35) has solutions. Now take
. Then it is clear that
verifies the requirement (a) of Lemma 2.11.
When ,
is a constant vector in
with
, from the definition of
, we can naturally derive
whether the system (3.35) has solutions or not. This shows that the condition (b) of Lemma 2.11 is satisfied.
Finally, we will prove that the condition (c) of Lemma 2.11 is valid. Define the homotopy by

where

where is a parameter. From (3.37), it is easy to show that
. Moreover, one can easily show that the algebraic equation

has a unique positive solution in
. Note that
(identical mapping), since
, according to the invariance property of homotopy, direct calculation produces

where is the Brouwer degree. By now we have proved that
verifies all requirements of Lemma 2.11. Therefore, (1.4) has at least one
-periodic solution in
. The proof is complete.
Corollary 3.2.
If the conditions in Theorem 3.1 hold, then both the corresponding continuous model (1.6) and the discrete model (1.7) have at least one -periodic solution.
Remark 3.3.
If and
in (1.6), then the system (1.6) reduces to the continuous ratio-dependence predator-prey diffusive system proposed in [17].
Remark 3.4.
If we only consider the prey population in one-patch environment and ignore the dispersal process in the system (1.4), then the classical ratio-dependence two species predator-prey model in particular of (1.4) with Michaelis-Menten functional response and time delay on time scales

where are positive
-periodic functions,
is nonnegative constant. It is easy to obtain the corresponding conclusions on time scales for the system (3.40).
Corollary 3.5.
Suppose that (i) , (ii)
hold, then (3.40) has at least one
-periodic solution.
Remark 3.6.
The result in Corollary 3.5 is same as those for the corresponding continuous and discrete systems.
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Acknowledgments
The author is very grateful to his supervisor Prof. M. Fan and the anonymous referees for their many valuable comments and suggestions which greatly improved the presentation of this paper. This work is supported by the Foundation for subjects development of Harbin University (no. HXK200716) and by the Foundation for Scientific Research Projects of Education Department of Hei-longjiang Province of China (no. 11513043).
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Liu, Z. Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion System on Time Scales. Adv Differ Equ 2009, 141589 (2009). https://doi.org/10.1155/2009/141589
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DOI: https://doi.org/10.1155/2009/141589
Keywords
- Periodic Solution
- Functional Response
- Jump Operator
- Continuation Theorem
- Fredholm Mapping