Persistence and global stability for n species ratio-dependent predator–prey system

Received 22 January 2001 Submitted by G.F. Webb
Abstract A n-species ratio-dependent predator–prey food-chain model with time delays is investigated. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are obtained for the global stability of the positive equilibrium of the system. 2002 Elsevier Science (USA). All rights reserved.
Keywords: Delay; Persiste Introduction The predator–prey system is an important population model and has been studied by many authors [1–3]. Standard Lotka–Volterra type models, on which a large body of existing predator–prey theory is built, assume that the per capita rate of predation depends on the prey numbers only. There is growing explicit biological and physiological evidence [4–7] that in many situations, especially when predators have to search, share or compete for food, a more suitable
* Corresponding author.
E-mail address: rxu88@ (R. Xu). 0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 2 4 7 X ( 0 2 ) 0 0 2 1 2 - 3
s ∈[−τ,0]
φ(s)
for φ ∈ C + .
Then, similar to the proof of Lemma 2.1 in Ref. [14], if we choose the initial function space of system (1.3) to be C + , it is easy to verify that, for any φ = (φ1 , φ2 , . . . , φn ) ∈ C + and φ(0) > 0, there exists a unique solution x(t, φ) of system (1.3) on [−τ, ∞) and remains positive for all t ∈ [0, ∞), such solutions of system (1.3) are called positive solution. Hence, in the rest of this paper, we always assume that φ ∈ C+, φ(0) > 0. (1.4)
Rui Xu a,∗ and Lansun Chen b
a Department of Mathematics, Shijiazhuang Mechanical Engineering College,
Shijiazhuang 050003, PR China
b Institute of Mathematics, Academia Sinica, Beijing 100080, PR China
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R. Xu, L. Chen / J. Math. Anal. Appl. 275 (2002) 27–43
general predator–prey model should be based on the “ratio-dependent” theory. This roughly states that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. Moreover, as the number of predators often changes slowly (relative to prey number), there is often competition among the predators, and the per capita rate of predation should therefore depend on the numbers of both prey and predator, most likely and simply on their ratio. These hypotheses are strongly supported by numerous field and laboratory experiment and observations [5,6,8,9]. A generic ratio-dependent predator–prey model takes the form x ˙ = xf (x) − yp(x/y), y ˙ = −dy + cyq(x/y), (1.1)
J. Math. Anal. Appl. 275 (2002) 27–43
Persistence and global stability for n-species ratio-dependent predator–prey system with time delays
where x, y respectively denote prey and predator density. The functions p(x) (the so-called predator functional response) and q(x) satisfy standard properties such as being non-negative and increasing, and equal to zero at zero. In particular, the ratio-dependent predator–prey model with Michaelis–Menten type functional response is as follows x ˙ = ax(1 − x/K) − cxy/(my + x), y ˙ = y −d + f x/(my + x) . (1.2)
On the other hand, it is generally recognized that some kinds of time delays are inevitable in population interactions and tend to be destabilizing in the sense that longer delays may destroy the stability of positive equilibria (see [10,11] and the references cited therein). Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. The effect of these kinds of delays on the asymptotic behavior of populations has been studied by a number of authors (see, for example, [12–15]). In this paper we incorporate time delays due to gestation into the ratiodependent predator–prey system. For the n-species ratio-dependent simple foodchain model with Michaelis–Menten type functional response, this results in the following delayed system a12 x2 (t) x ˙1 (t) = x1 (t) a1 − a11 x1 (t) − , m12 x2 (t) + x1 (t) ai,i −1 xi −1 (t − τi −1 ) x ˙i (t) = xi (t) −ai + m i −1,i xi (t − τi −1 ) + xi −1 (t − τi −1 ) (1.3) ai,i +1 xi +1 (t) − , i = 2, . . . , n − 1, mi,i +1 xi +1 (t) + xi (t) an,n−1 xn−1 (t − τn−1 ) x ˙n (t) = xn (t) −an + mn−1,n xn (t − τn−1 ) + xn−1 (t − τn−1 )