Dynamics of a stage-structured system with harvesting and discrete time delays

Abstract

A two-dimensional single-species stage-structured model is proposed and analyzed to study the effect of intraguild predation with harvesting of the adult species. Time lags in reproduction and maturation of the organism are introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. It is found that there are stability switches for time delays, and Hopf bifurcations when time delays cross through some critical values. The stability and direction of the Hopf bifurcations are determined by applying the normal form method and the center manifold theory. Computer simulations have been carried out to illustrate various analytical results.

Stage-Structured Cannibalism in a Ratio-Dependent System with Constant Prey Refuge and Harvesting of Matured Predator

“…In this paper, by introducing ratio-dependent functional response for the growth of Parrotfish and Pterois volitans and by considering refuge protection for Parrotfish, we obtain results that justify the real-field observations. The effect of cannibalism is not clearly stated in [20,21], whereas, in [22] it is observed that higher rate of cannibalism induces oscillation around the positive equilibrium. Since [22] is studied by means of a single-species stage-structured model, it could not be evaluated from this paper whether cannibalism of Pterois volitans actually reduces predation pressure on Parrotfish.…”

“…The effect of cannibalism is not clearly stated in [20,21], whereas, in [22] it is observed that higher rate of cannibalism induces oscillation around the positive equilibrium. Since [22] is studied by means of a single-species stage-structured model, it could not be evaluated from this paper whether cannibalism of Pterois volitans actually reduces predation pressure on Parrotfish. In this paper, we have considered the stage-structure of Pterois volitans with ratio-dependent functional response [23] under the assumption that matured Pterois volitans prey on Parrotfish and have reproductive ability.…”

“…Our mathematical model is a direct extension of the models studied in [20,21] under the assumption that algae is growing logistically and Parrotfish is growing on algae with a functional response which depends on the density of algae as well as Parrotfish. In our previous studies [22], it is observed that the system becomes oscillatory around the interior equilibrium when the invasion rate of Pterois volitans exceeds some certain critical threshold. But recent studies [2] indicate that due to the invasiveness of Pterois volitans, the herbivorous fish population drops drastically and as a result, due to scarcity of available foods, there is a loss of Pterois volitans species from that reef.…”

Potential Effects of Invasive Pterois Volitans in Coral Reefs

“…Almost all organisms have a life history that takes them through multiple stages from juvenile to adult (Zhang, Chen, & Neumann, 2000) and cannibalistic interactions are very common in stage-structured populations (Rudolf, 2007). To model the effect of invasive P. volitans, we have considered a two-stage-structured system (Bhattacharyya & Pal, 2014), with Parrotfish and P. volitans following a Holling type III functional response (Luwig, Jones, & Holling, 1978). This response function is sigmoid, rising slowly when resources are rare, accelerating when resources become more abundant and finally reaching a saturated upper limit (Edwards & Brindley, 1999).…”

Performance Analysis with Network-Enhanced Complexities: On Fading Measurements, Event-Triggered Mechanisms, and Cyber Attacks

“…Therefore, it is of great importance to understand how these challenging issues affect the closed-loop stability and performance properties of a controlled system or a filtering process. Fortunately, up to now, these network-related problems have gained much research effort and a wealth of literature has appeared on the related topics; see for example, [7][8][9][10][11][12][13][14][15][16][17] and the references therein. Roughly speaking, the published results have been concerned with the robust control and filtering problems of networked systems, H ∞ control and filtering problems, Kalman and extended Kalman filtering problems, optimal control problems, synchronization and consensus problems of complex networks, consensus and cooperative control problems of multiagent systems, and so on [18,19].…”