Estimates of Size of Cycle in a Predator-Prey System

Abstract

We consider a Rosenzweig-MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are also of independent interest.

A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model

“…For predator-prey models with eco-evolutionary dynamics in which ecological and evolutionary interactions occur on different time scales, relaxation oscillations were investigated by Piltz et al [21] and Shen et al [22]. For the classical predator-prey model with Monod functional response and small predator death rates, the unique periodic orbit, which was proved to exist by Liou and Cheng [18] (who corrected a flaw in the original proof of Cheng [3]) and Kuang and Freedman [14], was proved to form a relaxation oscillation by Hsu and Shi [9], Wang et al [24], and Lundström and Söderbacka [20], and the cyclicity of the limit cycle was investigated by Huzak [12]. For general functional responses, the number of relaxation oscillations depending on the number of local extrema of the prey-isocline was studied by Hsu [11].…”

Empirical parameterisation and dynamical analysis of the allometric Rosenzweig-MacArthur equations

“…Unfortunately early efforts to find analytic approximations of the oscillation amplitude of the Rosenzweig-MacArthur system have not been generalised [ 49 ]. Recent results have only been derived for specific—and restricted—parameter values [ 50 ]. However, the rescaled system (4) provides scope for us to examine the effects of perturbations in a simplified manner.…”

“…We propose that the interplay between scaling of ρ and predator-prey density slopes will have mathematically rich behaviours, especially as there is not yet a general solution for limit cycle amplitude in the Rosenzweig-MacArthur system [ 49 , 50 ], but it is beyond the scope of this work to interrogate this question in detail. Our empirically-driven paramaterisation for our model generates a theoretical size-abundance distribution matching the largest data study to date [ 27 ], and we note that to generate Damuth’s law [ 12 ], simply changing the scaling of one parameter b such that α b = 1/2 results in exponents of −0.81 and 3/4 for the size-abundance and predator-prey density scaling respectively.…”

“…We propose that the interplay between scaling of ρ and predator-prey density slopes will have mathematically rich behaviours, especially as there is not yet a general solution for limit cycle amplitude in the Rosenzweig-MacArthur system [49,50], but it is beyond the scope of this work to interrogate this question in detail. Our empirically-driven paramaterisation for We show examples of five predator sizes: 1E-6 (�), 1E-4(×), 1E-2 (+), 1E0 (4), and 1E2g ( � ).…”

Universal allometry from empirical parameters

“…Unfortunately early efforts to find analytic approximations of the oscillation amplitude of the Rosenzweig-Macarthur system have not been generalised [44]. Recent results have only been derived for specific -and restricted -parameter values [45]. However, the rescaled system (4) provides scope for us to examine the effects of perturbations in a simplified manner.…”