Effect of periodic environmental fluctuations on the Pearl–Verhulst model

We address the effect of periodic environmental fluctuations on the Pearl–Verhulst model in population dynamics and clarify several important issues very actively discussed in the recent papers by Lakshmi [Lakshmi BS. Oscillating population models. Chaos Solitons & Fractals 2003;16:183–6; Lakshmi BS. Population models with time dependent parameters. Chaos Solitons & Fractals 2005;26:719–21], Leach and Andriopoulos [Leach PGL, Andriopoulos K. An oscillatory population model. Chaos Solitons & Fractals 2004;22:1183–8], Swart and Murrell [Swart JH, Murrell HC. An oscillatory model revisited. Chaos Solitons & Fractals 2007;32:1325–7]. Firstly, we review general results regarding existence and properties of periodic solutions and examine existence of a unique positive asymptotically stable periodic solution of a non-autonomous logistic differential equation when r(t)>0r(t)>0. Proceeding to the case where r(t)r(t) is allowed to take on negative values, we consider a modified Pearl–Verhulst equation because, as emphasized by Hallam and Clark [Hallam TG, Clark CE. Non-autonomous logistic equations as models of populations in deteriorating environment. J Theor Biol 1981;93:303–11], use of the classic one leads to paradoxical biological conclusions. For a modified logistic equation with ωω-periodic coefficients, we establish existence of a unique asymptotically stable positive periodic solution with the same period. Special attention is paid to important cases where time average of the intrinsic growth rate is non-positive. Results of computer simulation demonstrating advantages of a modified equation for modeling periodic environmental fluctuations are presented.