A predator-prey model with optimal suppression of reproduction in the prey
The main objective of this paper is to formulate and analyze a predator-prey model with the possibility of suppressed reproduction in the prey. We commence with a single-species model in which we assume a trade-off between the reproductive effort and competitive ability. If no behavioral adjustment of the rate of reproduction is allowed for, the type with the reproductive rate maximizing the equilibrium population size prevails, regardless of whether the equilibrium point is stable or not. However, when behavioral adjustments are allowed for, the winning type suppresses reproduction, to some extent, whenever the equilibrium is unstable, and this behavior decreases the mean amplitude of population oscillations. We then extend this model to include predation, assuming a trade-off between the reproductive effort and an individual's ability to avoid predation. Optimal adjustment of reproduction by the prey adds to population stability, as measured by the mean amplitude of oscillations. Our paper is organized as follows. In Section 2 we introduce the notions of fixed and adjustable strategy in the single-species case. We find the best fixed strategy and investigate the asymptotic behavior of solutions to a model describing competition between fixed and adjustable strategies. In Section 3 we introduce a new discrete predator-prey model, which can be regarded as a combination of the \textit{W. E. Ricker} model [J. Fish. Res. Board Can. 11, No. 5, 559-623 (1954)], the \textit{A. J. Nicholson} and \textit{V. A. Bailey} model [Zool. Soc. (Lond.) Proc. 3, 551-598 (1935)], and the continuous Lotka-Volterra model. A mechanistic explanation of the model is included in Section 3. In Section 4 we investigate the dynamics of this discrete predator-prey model, and this study culminates in an important stability theorem. In Section 5 we investigate the dynamical consequences of the optimal suppression of reproduction in the prey. The analytical study of the predator-prey model is supplemented with a numerical study in Section 6. In Section 7 we relate our results to those of other authors and discuss their biological implications. The proofs of the theorems and propositions can be found in Section 8.
scientific article; zbMATH DE number 897910
scientific article