Stability of a delayed competitive model with saturation effect and interval biological parameters
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Stability of a delayed competitive model with saturation effect and interval biological parameters Siyu Chen1 · Zhijun Liu1
· Lianwen Wang1 · Jing Hu1
Received: 20 February 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract This work presents a delayed two-species competitive model with interval biological parameters, in which each interspecific competition term suffers delay and saturation effect. Sufficient criteria for the existence, local and global asymptotic stability of the unique positive equilibrium are established, respectively. Our theoretical and simulated results show that under appropriate conditions the fuzziness of biological parameters plays a critical role in the stability properties of the system while time delays have little influence. Keywords Competition model with saturation effect · Delay · Interval number · Stability Mathematics Subject Classification 92D25 · 34K20 · 34D23
1 Introduction Competition phenomena are always interesting issues on population dynamics which receive great attention owing to their universal existence and significance. Understanding the nature of competitive interaction is important in maintaining balance between different species. Lots of deterministic or stochastic competitive models have been proposed and investigated in the last few decades, and there exist extensive literatures concerned with their dynamics, for example, stability and bifurcation [1–4], periodic
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Zhijun Liu [email protected]; [email protected] Siyu Chen [email protected] Lianwen Wang [email protected] Jing Hu [email protected]
1
Department of Mathematics, Hubei Minzu University, Enshi 445000, Hubei, People’s Republic of China
123
S. Chen et al.
solution [5–11], almost periodic solution [12,13], persistence and extinction [14–19], stationary distribution [20,21] and so on. In particular, Gopalsamy [1] formulated a two-species competitive model with saturation effect as follows ⎧ b1 z 2 dz 1 ⎪ ⎨ = z 1 r 1 − a1 z 1 − , dt 1 + z2 (1.1) dz b z ⎪ ⎩ 2 = z 2 r 2 − a2 z 2 − 2 1 , dt 1 + z1 where z i (t)(i = 1, 2) is the density of the ith species at time t. ri > 0, ai > 0 are respectively the intrinsic growth rate and the intraspecific competing rate of the ith species, and b1 , b2 are the interspecific competing rates. Obviously, each species is governed by the famous logistic equation in the absence of interspecific interactions. In fact, the interaction term bi z j /(1 + z j ) is associated with an increasing function and has a saturation value bi for sufficiently large z j (i, j = 1, 2; i = j). For the biological meanings and relevant investigations of system (1.1) or similar to system (1.1) we refer the readers to [1,3,4,11,13,15,18,19] . In the following, with the idea of delay effect we further consider a delayed version ⎧ dz 1 ⎪ ⎨ = z 1 r 1 − a1 z 1 − dt dz 2 ⎪ ⎩ = z 2 r 2 − a2 z 2 − dt
b1 z 2 (t − τ1 ) , 1 + z 2 (t − τ1 ) b2 z 1 (t − τ2 ) , 1 + z 1 (t − τ2 )
(1.2)
from which we can see that the
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