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Multiple positive periodic solutions for a foodlimited twospecies GilpinAyala competition patch system with periodic harvesting terms
Journal of Inequalities and Applications volume 2012, Article number: 291 (2012)
Abstract
By using Mawhin’s coincidence degree theory and some inequality techniques, this paper establishes a new sufficient condition on the existence of at least eight positive periodic solutions for a foodlimited twospecies GilpinAyala competition patch system with periodic harvesting terms. An example is given to illustrate the effectiveness of the result.
1 Introduction
In the past years, the study of population dynamics with harvesting in mathematical bioeconomics, due to its theoretical and practical significance in the optimal management of renewable resources, has attracted much attention [1–8]. Huusko and Hyvarinen in [9] pointed out that ‘the dynamics of exploited populations are clearly affected by recruitment and harvesting, and the changes in harvesting induced a tendency to generation cycling in the dynamics of a freshwater fish population.’ Recently, some researchers have paid much attention to the investigation of harvestinginduced multiple positive periodic solutions for some population systems under the assumption of periodicity of the parameters by using Mawhin’s coincidence degree theory [5–8]. In 1973, Gilpin and Ayala in [10] firstly proposed and studied a few GilpinAyala type competition models. Since then, many papers have been published on the dynamics of GilpinAyala type competition models (for example, see [11–15]).
In this paper, we consider a foodlimited twospecies GilpinAyala competition patch system with harvesting terms:
where {x}_{1} and y are the population densities of species x and y in patch 1, and {x}_{2} is the density of species x in patch 2. Species y is confined to patch 1, while species x can diffuse between two patches due to the spatial heterogeneity and unbalanced food resources. {D}_{i}(t) (i=1,2) are diffusion coefficients of species x. {a}_{1}(t) ({a}_{2}(t)) is the natural growth rate of species x in patch 1 (patch 2), {a}_{3}(t) is the natural growth rate of species y, {a}_{13}(t), {a}_{31}(t) are the interspecies competition coefficients. {a}_{ii}(t) (i=1,2,3) are the densitydependent coefficients. {k}_{i}(t) (i=1,2) are the population numbers of species x at saturation in patch 1 (patch 2), and {k}_{3}(t) is the population number of species y at saturation in patch 1, respectively. {H}_{i}(t) (i=1,2,3) denote the harvesting rates. {\theta}_{i} (i=1,2,3) represent a nonlinear measure of interspecific interference. When {c}_{i}(t)\ne 0 (i=1,2,3), \frac{{a}_{i}(t)}{{k}_{i}(t){c}_{i}(t)} (i=1,2,3) are the rate of replacement of mass in the population at saturation (including the replacement of metabolic loss and of dead organisms). In this case, system (1.1) is a foodlimited population model. For other foodlimited population models, we refer to [16–19].
To our knowledge, few papers have been published on the existence of multiple positive periodic solutions for GilpinAyala type competition patch models. Motivated by the work of Chen [20], we study the existence of multiple positive periodic solutions of (1.1) by using Mawhin’s coincidence degree theory. Since system (1.1) involves the diffusion terms, the rates of replacement and the interspecific interference, the methods used in [5–8] are not available to system (1.1).
2 Existence of multiple positive periodic solutions
For the sake of convenience and simplicity, we denote
where g is a nonnegative continuous Tperiodic function.
Set
From now on, we always assume that
(H_{1}) {k}_{i}(t), {a}_{i}(t), {a}_{ii}(t), {H}_{i}(t), {c}_{i}(t) (i=1,2,3), {a}_{13}(t), {a}_{31}(t), {D}_{i}(t) (i=1,2) are positive continuous Tperiodic functions. {\theta}_{i} (i=1,2,3) are positive constants.
(H_{2}) \frac{{k}_{1}^{l}}{{k}_{1}^{l}+{c}_{1}^{u}{N}_{1}}{(\frac{{a}_{1}}{{k}_{1}})}^{l}>{(\frac{{a}_{13}}{{k}_{1}})}^{u}{(\frac{{a}_{3}}{{a}_{33}})}^{u}+{D}_{1}^{u}+(1+{\theta}_{1}){[{(\frac{{a}_{11}}{{k}_{1}})}^{u}]}^{\frac{1}{1+{\theta}_{1}}}{[\frac{{H}_{1}^{u}}{{\theta}_{1}}]}^{\frac{{\theta}_{1}}{1+{\theta}_{1}}}.
(H_{3}) \frac{{k}_{2}^{l}}{{k}_{2}^{l}+{c}_{2}^{u}{N}_{1}}{(\frac{{a}_{2}}{{k}_{2}})}^{l}>{D}_{2}^{u}+(1+{\theta}_{2}){[{(\frac{{a}_{22}}{{k}_{2}})}^{u}]}^{\frac{1}{1+{\theta}_{2}}}{[\frac{{H}_{2}^{u}}{{\theta}_{2}}]}^{\frac{{\theta}_{2}}{1+{\theta}_{2}}}.
(H_{4}) \frac{{k}_{3}^{l}}{{k}_{3}^{l}+{c}_{3}^{u}{N}_{2}}{(\frac{{a}_{3}}{{k}_{3}})}^{l}>{(\frac{{a}_{31}}{{k}_{3}})}^{u}{N}_{1}^{{\theta}_{1}}+(1+{\theta}_{3}){[{(\frac{{a}_{33}}{{k}_{3}})}^{u}]}^{\frac{1}{1+{\theta}_{3}}}{[\frac{{H}_{3}^{u}}{{\theta}_{3}}]}^{\frac{{\theta}_{3}}{1+{\theta}_{3}}}.
(H_{5}) {H}_{i}^{l}>{D}_{i}^{u}{N}_{1} (i=1,2).
We first make the following preparations [21].
Let X, Z be normed vector spaces, L:domL\subset X\to Z be a linear mapping, N:X\times [0,1]\to Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL=codimImL<+\mathrm{\infty} and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors P:X\to X and Q:Z\to Z such that ImP=KerL, ImL=KerQ=Im(IQ). If we define {L}_{P}:domL\cap KerP\to ImL as the restriction L{}_{domL\phantom{\rule{0.2em}{0ex}}\cap \phantom{\rule{0.2em}{0ex}}KerP} of L to domL\cap KerP, then {L}_{P} is invertible. We denote the inverse of that map by {K}_{P}. If Ω is an open bounded subset of X, the mapping N will be called Lcompact on \overline{\mathrm{\Omega}}\times [0,1] if QN(\overline{\mathrm{\Omega}}\times [0,1]) is bounded and {K}_{P}(IQ)N:\overline{\mathrm{\Omega}}\times [0,1]\to X is compact, i.e., continuous and such that {K}_{P}(IQ)N(\overline{\mathrm{\Omega}}\times [0,1]) is relatively compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ\to KerL.
For convenience, we introduce Mawhin’s continuation theorem [[21], p.29] as follows.
Lemma 2.1 Let L be a Fredholm mapping of index zero and let N:\overline{\mathrm{\Omega}}\times [0,1]\to Z be Lcompact on \overline{\mathrm{\Omega}}\times [0,1]. Suppose

(a)
Lu\ne \lambda N(u,\lambda ) for every u\in domL\cap \partial \mathrm{\Omega} and every \lambda \in (0,1);

(b)
QN(u,0)\ne 0 for every u\in \partial \mathrm{\Omega}\cap KerL;

(c)
Brouwer degree {deg}_{B}(JQN(\cdot ,0){}_{KerL},\mathrm{\Omega}\cap KerL,0)\ne 0.
Then Lu=N(u,1) has at least one solution in domL\cap \overline{\mathrm{\Omega}}.
Set
Lemma 2.2 Assume that a, b, c, α are positive constants and
Then there exist 0<{x}^{}<{x}^{+} such that
Proof Since
implies that
we have
From this, it is easy to see that the assertion holds.
Set
□
Lemma 2.3 Assume that (H_{1})(H_{5}) hold. Then the following assertions hold:

(1)
There exist 0<{u}_{i}^{}<{u}_{i}^{+} such that
{M}_{i}\left({u}_{i}^{}\right)={M}_{i}\left({u}_{i}^{+}\right)=0,
and

(2)
There exist 0<{x}_{i}^{}<{x}_{i}^{+} such that
{p}_{i}\left({x}_{i}^{}\right)={p}_{i}\left({x}_{i}^{+}\right)=0,
and

(3)
There exist 0<{l}_{i}^{}<{l}_{i}^{+} such that
{m}_{i}\left({l}_{i}^{}\right)={m}_{i}\left({l}_{i}^{+}\right)=0,
and
(4)
Proof It follows from (H_{1})(H_{5}) and Lemma 2.2 that the assertions (1)(3) hold. Noticing that
we have
It follows from this and the assertions (1)(3) that the assertion (4) also holds. □
Lemma 2.4 [22]
Assume that x\ge 0, y\ge 0, p>1, q>1, and \frac{1}{p}+\frac{1}{q}=1. Then the following inequality holds:
Now, we are ready to state the following main result of this paper.
Theorem 2.1 Assume that (H_{1})(H_{5}) hold. Then system (1.1) has at least eight positive Tperiodic solutions.
Proof Since we are concerned with positive solutions of (1.1), we make the change of variables
Then (1.1) is rewritten as
Take
and define
Equipped with the above norm \parallel \cdot \parallel, it is easy to verify that X and Z are Banach spaces.
Set
For any u\in X, because of the periodicity, we can easily check that {\mathrm{\Delta}}_{i}(u,t,\lambda )\in C({R}^{2},R) (i=1,2,3) are Tperiodic in t.
Let
Here, for any k\in {R}^{3}, we also identify it as the constant function in X or Z with the constant value k. It is easy to see that
is closed in Z, dimKerL=codimImL=3, and P, Q are continuous projectors such that
Therefore, L is a Fredholm mapping of index zero. On the other hand, {K}_{p}:ImL\mapsto domL\cap KerP has the form
Thus,
where
Obviously, QN and {K}_{p}(IQ)N are continuous. By the ArzelaAscoli theorem, it is not difficult to show that \overline{{K}_{p}(IQ)N(\overline{\mathrm{\Omega}}\times [0,1]}) is compact for any open bounded set \mathrm{\Omega}\subset X. Moreover, QN(\overline{\mathrm{\Omega}}\times [0,1]) is bounded. Thus, N is Lcompact on \overline{\mathrm{\Omega}}\times [0,1] with any open bounded set \mathrm{\Omega}\subset X.
In order to apply Lemma 2.1, we need to find eight appropriate open, bounded subsets {\mathrm{\Omega}}_{i} (i=1,2,\dots ,8) in X.
Corresponding to the operator equation Lu=\lambda N(u,\lambda ), \lambda \in (0,1), we have
Suppose that {({u}_{1}(t),{u}_{2}(t),{u}_{3}(t))}^{T} is a Tperiodic solution of (2.3), (2.4) and (2.5) for some \lambda \in (0,1).
Choose {t}_{i}^{M},{t}_{i}^{m}\in [0,T], i=1,2,3, such that
Then it is clear that
From this and (2.3), (2.4), (2.5), we obtain that
and
Claim A.
and
For u({t}_{i}^{M}) (i=1,2), there are two cases to consider.
Case 1. Assume that {u}_{1}({t}_{1}^{M})\ge {u}_{2}({t}_{2}^{M}), then {u}_{1}({t}_{1}^{M})\ge {u}_{2}({t}_{1}^{M}).
From this and (2.6), we have
which implies
That is,
Case 2. Assume that {u}_{1}({t}_{1}^{M})<{u}_{2}({t}_{2}^{M}), then {u}_{2}({t}_{2}^{M})>{u}_{1}({t}_{2}^{M}).
From this and (2.7), we have
which implies
That is,
Therefore,
For {u}_{3}({t}_{3}^{M}), it follows from (2.8) that
which implies
Claim B.
and
It follows from (2.6) that
Therefore,
From this and noticing that
we have
which implies
From the assertion (1) of Lemma 2.3 and the above inequality, we have
Similarly, from (2.9), we obtain
By a similar argument, it follows from (2.7) that
From the assertion (1) of Lemma 2.3 and the above inequality, we have
Similarly, from (2.10), we obtain
By a similar argument, it follows from (2.8) and (2.12) that
From the assertion (1) of Lemma 2.3 and the above inequality, we have
Similarly, from (2.11), we obtain
Claim C.
and
It follows from (2.6) that
Hence, we have
From the assertion (3) of Lemma 2.3 and the above inequality, we have
Similarly, from (2.9), we obtain
By a similar argument, it follows from (2.7) that
From the assertion (3) of Lemma 2.3 and the above inequality, we have
Similarly, from (2.10), we obtain
By a similar argument, it follows from (2.8) that
From the assertion (3) of Lemma 2.3 and the above inequality, we have
Similarly, from (2.11), we obtain
It follows from (2.14), (2.15), (2.20), (2.21) that
It follows from (2.16), (2.17), (2.22), (2.23) that
It follows from (2.18), (2.19), (2.24), (2.25) that
Clearly, {l}_{i}^{\pm}, {u}_{i}^{\pm} (i=1,2,3) are independent of λ. Now, let us consider QN(u,0) with u={({u}_{1},{u}_{2},{u}_{3})}^{T}\in {R}^{3}. Note that
Letting QN(u,0)=0, we have
Therefore, it follows from the assertion (2) of Lemma 2.3 that QN(u,0)=0 has eight distinct solutions:
Let
Then {\mathrm{\Omega}}_{1},{\mathrm{\Omega}}_{2},\dots ,{\mathrm{\Omega}}_{8} are bounded open subsets of X. It follows from (2.1) and (2.35)(2.38) that {\tilde{u}}_{i}\in {\mathrm{\Omega}}_{i} (i=1,2,\dots ,8). From (2.1), (2.26)(2.31), it is easy to see that {\overline{\mathrm{\Omega}}}_{i}\cap {\overline{\mathrm{\Omega}}}_{j}=\mathrm{\varnothing} (i,j=1,2,\dots ,8, i\ne j) and {\mathrm{\Omega}}_{i} satisfies (a) in Lemma 2.1 for i=1,2,\dots ,8. Moreover, QN(u,0)\ne 0 for u\in \partial {\mathrm{\Omega}}_{i}\cap KerL. By Lemma 2.2, a direct computation gives
Here, J is taken as the identity mapping since ImQ=KerL. So far we have proved that {\mathrm{\Omega}}_{i} satisfies all the assumptions in Lemma 2.1. Hence, (2.2) has at least eight Tperiodic solutions {({u}_{1}^{i}(t),{u}_{2}^{i}(t),{u}_{3}^{i}(t))}^{T} (i=1,2,\dots ,8) and {({u}_{1}^{i},{u}_{2}^{i},{u}_{3}^{i})}^{T}\in domL\cap {\overline{\mathrm{\Omega}}}_{i}. Obviously, {({u}_{1}^{i},{u}_{2}^{i},{u}_{3}^{i})}^{T} (i=1,2,\dots ,8) are different. Let {x}_{j}^{i}(t)={e}^{{u}_{j}^{i}(t)} (j=1,2), {y}^{i}(t)={e}^{{u}_{3}^{i}(t)} (i=1,2,\dots ,8). Then {({x}_{1}^{i}(t),{x}_{2}^{i}(t),{y}^{i}(t))}^{T} (i=1,2,\dots ,8) are eight different positive Tperiodic solutions of (1.1). The proof is complete. □
Corollary 2.1 In addition to (H_{1}), (H_{5}), assume further that the following conditions hold:
(H_{2})^{∗} \frac{{k}_{1}^{l}}{{k}_{1}^{l}+{c}_{1}^{u}{N}_{1}}{(\frac{{a}_{1}}{{k}_{1}})}^{l}>{(\frac{{a}_{13}}{{k}_{1}})}^{u}{(\frac{{a}_{3}}{{a}_{33}})}^{u}+{D}_{1}^{u}+{(\frac{{a}_{11}}{{k}_{1}})}^{u}+{H}_{1}^{u}.
(H_{3})^{∗} \frac{{k}_{2}^{l}}{{k}_{2}^{l}+{c}_{2}^{u}{N}_{1}}{(\frac{{a}_{2}}{{k}_{2}})}^{l}>{D}_{2}^{u}+{(\frac{{a}_{22}}{{k}_{2}})}^{u}+{H}_{2}^{u}.
(H_{4})^{∗} \frac{{k}_{3}^{l}}{{k}_{3}^{l}+{c}_{3}^{u}{N}_{2}}{(\frac{{a}_{3}}{{k}_{3}})}^{l}>{(\frac{{a}_{31}}{{k}_{3}})}^{u}{N}_{1}^{{\theta}_{1}}+{(\frac{{a}_{33}}{{k}_{3}})}^{u}+{H}_{3}^{u}.
Then system (1.1) has at least eight positive Tperiodic solutions.
Proof By Lemma 2.4, we have
Therefore, the conditions in Theorem 2.1 are satisfied. □
Example 2.2 In (1.1), take
Then we have
Therefore,
Hence, the conditions in Corollary 2.1 are satisfied. By Corollary 2.1, system (1.1) has at least eight positive fourperiodic solutions.
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Acknowledgements
This research is supported by the National Natural Science Foundation of China (Grant No. 10971085).
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Fang, H. Multiple positive periodic solutions for a foodlimited twospecies GilpinAyala competition patch system with periodic harvesting terms. J Inequal Appl 2012, 291 (2012). https://doi.org/10.1186/1029242X2012291
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DOI: https://doi.org/10.1186/1029242X2012291