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 and Fractals 2003;16:183-6; Lakshmi BS. Population models with time dependent parameters. Chaos Solitons and Fractals 2005;26:719-21], Leach and Andriopoulos [Leach PGL, Andriopoulos K. An oscillatory population model. Chaos Solitons and Fractals 2004;22:1183-8], Swart and Murrell [Swart JH, Murrell HC. An oscillatory model revisited. Chaos Solitons and 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)>0. Proceeding to the case where 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 ω-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.