Permanence of a Discrete Model of Mutualism with Infinite Deviating Arguments

Volume 2010, Article ID 931798,7pages doi:10.1155/2010/931798

Research Article

Permanence of a Discrete Model of Mutualism with Infinite Deviating Arguments

Xuepeng Li and Wensheng Yang

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350007, China

Correspondence should be addressed to Wensheng Yang,[email protected] Received 15 July 2009; Accepted 13 January 2010

Academic Editor: Binggen Zhang

Copyrightq2010 X. Li and W. Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a discrete model of mutualism with infinite deviating arguments, that isx1n1 x1nexp{r1nK1nα1n_∞

s0J2sx2n−s/1_∞

s0J2sx2n−s−x1n−σ1n}, x2n 1 x2nexp{r2nK2n α2n_∞

s0J1sx1n−s/1_∞

s0J1sx1n−s−x2n−σ2n}.

By some Lemmas, sufficient conditions are obtained for the permanence of the system.

1. Introduction

Chen and You1studied the following two species integro-differential model of mutualism:

dN1t

dt r1tN1t

K1t α1t_∞

0 J2sN2t−sds 1_∞

0 J2sN2t−sds −N1t−σ1t

,

dN2t

dt r2tN2t

K₂t α₂t_∞

0 J₁sN1t−sds 1_∞

0 J1sN1t−sds −N2t−σ2t

,

1.1

whereri, Ki, αi, andσi, i1,2 are continuous functions bounded above and below by positive constants: a_i > K_i, i 1,2; J_i ∈ C0,∞,0,∞ and _∞

0J_isds 1, i 1,2.Using the differential inequality theory, they obtained a set of sufficient conditions to ensure the permanence of system1.1. For more background and biological adjustments of system1.1, one could refer to1–4and the references cited therein.

However, many authors5–12have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Also, since discrete time models can also provide efficient

computational models of continuous models for numerical simulations, it is reasonable to study discrete time models governed by difference equations. Another permanence is one of the most important topics on the study of population dynamics. One of the most interesting questions in mathematical biology concerns the survival of species in ecological models. It is reasonable to ask for conditions under which the system is permanent.

Motivated by the above question, we consider the permanence of the following discrete model of mutualism with infinite deviating arguments:

x₁n1 x₁nexp

r₁n

K₁n α₁n_∞

s0J₂sx2n−s 1_∞

s0J₂sx2n−s −x₁n−σ₁n ,

x2n1 x2nexp

r2n

K₂n α₂n_∞

s0J₁sx1n−s 1_∞

s0J₁sx1n−s −x2n−σ2n ,

1.2

where xin, i 1,2 is the density of mutualism species i at the nth generation. For {rin},{Kin},{αin},{Jin}, and{σin}, i 1,2 are bounded nonnegative sequences such that

0< r_i^l≤r_i^u, 0< α^l_i≤α^u_i, 0< K_i^l≤K_i^u, 0< σ_i^l≤σ_i^u, ∞ n0

Jin 1. 1.3

Here, for any bounded sequence{an},a^usup_n∈Nan, a^linf_n∈Nan.

Let σ sup_n{σin, i 1,2},we consider1.2 together with the following initial condition:

xiθ ϕiθ≥0, θ∈N−τ,0 {−τ,−τ1, . . . ,0}, ϕi0>0. 1.4 It is not difficult to see that solutions of1.2and1.4are well defined for alln≥ 0 and satisfy

xin>0, for n∈Z, i1,2. 1.5 The aim of this paper is, by applying the comparison theorem of difference equation and some lemmas, to obtain a set of sufficient conditions which guarantee the permanence of system1.2.

2. Permanence

In this section, we establish permanence results for system1.2.

Following Comparison Theorem of difference equation is Theorem2.6 of 13, page 241.

Lemma 2.1. Letk ∈ N_k

0 {k0, k₀ 1, . . . , k₀l, . . .}, r ≥ 0. For any fixedk, gk, r is a non- decreasing function with respect to r, and for k ≥ k0, following inequalities hold: yk 1 ≤ gk, yk, uk1≥gk, uk.Ifyk0≤uk0, thenyk≤ukfor allk≥k₀.

Now let us consider the following single species discrete model:

Nk1 Nkexp{ak−bkNk}, 2.1

where{ak}and{bk}are strictly positive sequences of real numbers defined fork∈N {0,1,2, . . .}and 0< a^l≤a^u,0< b^l ≤b^u. Similar to the proof of Propositions 1 and 3 in6, we can obtain the following.

Lemma 2.2. Any solution of system2.1with initial conditionN0>0 satisfies m≤ lim

k→∞infNk≤ lim

k→∞supNk≤M, 2.2

where

M 1

b^l exp{a^u−1}, m a^l b^u exp

a^l−b^uM

. 2.3

Lemma 2.3see14. Letxnand bnbe nonnegative sequences defined onN, andc ≥ 0 is a constant. If

xn≤cⁿ⁻¹

s0

bsxs, forn∈N, 2.4

then

xn≤c n−1

s0

1bs, forn∈N. 2.5

Lemma 2.4see2. Letx:Z → Rbe a nonnegative bounded sequences, and letH:N → Rbe a nonnegative sequence such that_∞

n0J_in 1. Then

nlim→∞infxn≤ lim

n→∞inf n

s−∞Hn−sxs

≤ lim

n→∞sup

n s−∞

Hn−sxs≤ lim

n→∞supxn.

2.6

Proposition 2.5. Letx1n, x2nbe any positive solution of system1.2, then

nlim→∞supx_in≤M_i, i1,2, 2.7

where

Miexp 2r_i^u

K_i^uα^u_i

, i1,2. 2.8

Proof. Letx1n, x2nbe any positive solution of system1.2, then from the first equation of system1.2we have

x1n1≤x1nexp

r1n

K₁n α₁n_∞

s0J₂sx2n−s 1_∞

s0J₂sx2n−s x₁nexp

r₁n

K₁n 1_∞

s0J₂sx2n−sα1n_∞

s0J2sx2n−s 1_∞

s0J₂sx2n−s

≤x₁nexp

r₁n K₁n

1 α₁n_∞

s0J₂sx2n−s _∞

s0J2sx2n−s x₁nexp{r1nK1n α₁n}

≤x1nexp r₁^u

K₁^uα^u₁ .

2.9

Letx1n exp{u1n}, then

u₁n1≤u₁n r₁^u

K₁^uα^u₁ r₁^u

K₁^uα^u₁ ⁿ

s0

bsxs, 2.10

where

bs

⎧⎨

⎩

0, 0≤s≤n−1,

1, sn. 2.11

Whenu₁nis nonnegative sequence, by applying Lemma2.3, it immediately follows that

u₁n1≤2r₁^u

K^u₁α^u₁

. 2.12

Whenu1nis negative sequence,2.12also holds. From2.12, we have

nlim→∞supx₁n≤exp 2r₁^u

K₁^uα^u₁

:M₁. 2.13

By using the second equation of system1.2, similar to the above analysis, we can obtain

nlim→∞supx2n≤exp 2r₂^u

K₂^uα^u₂

:M2. 2.14

This completes the proof of Proposition2.5.

Now we are in the position of stating the permanence of system1.2.

Theorem 2.6. Under the assumption1.3, system 1.2 is permanent, that is, there exist positive constantsm_i, M_i, i 1,2 which are independent of the solutions of system1.2such that, for any positive solutionx1n, x2nof system1.2with initial condition1.4, one has

m_i≤ lim

n→∞infx_in≤ lim

n→∞supx_in≤M_i, i1,2. 2.15

Proof. By applying Proposition2.5, we see that to end the proof of Theorem2.6it is enough to show that under the conditions of Theorem2.6

nlim→∞infx_in≥m_i. 2.16

From Proposition2.5, For allε >0, there exists aN₁>0, N₁∈N,For alln > N₁,

x_in≤M_iε. 2.17

According to Lemma2.4, from2.13and2.14we have

nlim→∞sup ∞ s0

Jisxin−s lim

n→∞sup

n k−∞

Jin−kxik≤Mi, i1,2. 2.18

For aboveε >0, according to2.18, there exists a positive integerN2, such that, for alln > N2, ∞

s0

Jisxin−s≤Miε, i1,2. 2.19

Thus, for alln > max{N1, N₂}σ, from the first equation of system1.2, it follows that

x₁n1≥x₁nexp

r₁n

K^l₁

1 M2ε−M1ε

≥x1nexp

r₁^lK₁^l

1 M2ε−r₁^uM1ε

.

2.20

It follows that, forn≥σ₁n, n−1

in−σ1n

x1i1≥ ⁿ⁻¹

in−σ1n

x1iexp

r₁^lK₁^l

1 M₂ε−r₁^uM1ε

. 2.21

Hence

x1n≥x1n−σ1nexp

r₁^lK^l₁

1 M₂εσ₁^l −r₁^uM1εσ₁^u

. 2.22

In other words,

x1n−σ1n≤x1nexp

− r₁^lK₁^l

1 M₂εσ₁^lr₁^uM1εσ₁^u

. 2.23

From the first equation of system1.2and2.23, for alln > max{N1, N₂}σ, it follows that

x₁n1≥x₁nexp

− r₁^lK^l₁

1 M₂ε−r₁^uexp

− r₁^lK^l₁

1 M₂εσ₁^lr₁^uM1εσ₁^u

x₁n

. 2.24 By applying Lemmas2.1and2.2to2.24, it immediately follows that

nlim→∞infx1n≥ r₁^lK₁^l

r₁^u1 M₂εexp

r₁^lK₁^l

1 M₂εσ₁^l−r₁^uM1εσ₁^u

×exp

r₁^lK^l₁

1 M₂ε−r₁^uexp

− r₁^lK^l₁

1 M₂εσ₁^l r₁^uM1εσ₁^u

M1

. 2.25 Settingε → 0, it follows that

nlim→∞infx₁n≥ r₁^lK^l₁ r₁^u1M2exp

r₁^lK₁^l 1M2

σ₁^l −r₁^uM₁σ₁^u

×exp

r₁^lK^l₁

1M2 −r₁^uexp

− r₁^lK^l₁

1M2σ₁^lr₁^uM1σ₁^u

M1

.

2.26

Similar to the above analysis, from the second equation of system1.2, we have that

nlim→∞infx2n≥ r₂^lK^l₂ r₂^u1M₁exp

r₂^lK₂^l

1M₁σ₂^l −r₂^uM2σ₂^u

×exp

r₂^lK^l₂

1M₁ −r₂^uexp

− r₂^lK^l₂

1M₁σ₂^lr₂^uM2σ₂^u

M2

.

2.27

This completes the proof of Theorem2.6.

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