Full Delay Logistic Population Model with Sustainable Harvesting

Abstract

The paper studies the dynamics of a full delay-logistic population model incorporated with a proportionate harvesting function. The study discusses the stability of the model in comparison with the well-known Hutchinson logistic growth equation with harvesting function using the rate of harvesting as a bifurcation parameter to determine sustainable harvesting rate even at a bigger time delay of τ = 3.00 . In all cases, the Hutchinson equation with harvesting was forced to converge to equilibrium using an additional and a different time delay parameter, a deficiency previous researchers have failed to address when the Hutchinson model is used for this purpose. The population fluctuations are catered for with this model making the estimated maximum sustainable growth and harvest reflect realities as this model drastically reduces time-delay associated oscillations compared to the well-known Hutchinson delayed logistic models. The numerical simulations were be done using the MatLab Software.

Stability scenarios and period‐doubling onset of chaos in a population model with delayed harvesting

“…Toaha [1] studied the effects of delayed harvesting on the dynamics of the logistic equation for single‐species population and observed a stability change (stable to unstable only) due to the time delay for fixed harvesting effort. Similar stability scenario was also reported recently for a fully delayed logistic equation incorporating a delayed harvesting [2] and a weighted average harvesting [3]. The stability of a delayed Nicholson blowflies model under a delayed population harvesting strategy was explored by Liu and Wei [4].…”

“…Case VI: A two-fold positive 𝜔 exist When h = h 2 (i.e., b 2 1 − a 2 1 = 0), we have, as before, a two-fold positive 𝜔 given by 𝜔 = 𝜔 0 = √ bx * 𝑦 * (1+x * ) 3 . For the threshold delays, the second equation of (8) gives…”

“…It is a general curiosity to know whether the same scenario can occur for delayed harvesting. With this motivation, we shall now explore how the delayed harvesting could affect the stability behavior of the predator-prey system (2). We study the stability of the system around the predator-prey coexisting equilibrium E * only.…”

“…Although the effects of non-delayed harvesting on predator-prey interactions have been rigorously investigated, to the authors' knowledge, only a few literatures are available on the impacts of delayed harvesting in single-species [1][2][3][4] and multi-species [5][6][7][8][9] population dynamics. Toaha [1] studied the effects of delayed harvesting on the dynamics of the logistic equation for single-species population and observed a stability change (stable to unstable only) due to the time delay for fixed harvesting effort.…”