Oscillations in Age-Structured Models

In consumer-resource interactions, a resource is regarded as a biotic population that helps to maintain the population growth of its consumer, whereas a consumer exploits a resource and then reduces its growth rate. Bi-directional consumer-resource interactions describe the cases where each species acts as both a consumer and a resource of the other, which is the basis of many mutualisms. In uni-directional consumer-resource inter-actions one species acts as a consumer and the other as a material and/or energy resource while neither acts as both. In this paper we consider an age-structured model for uni-directional consumer-resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the con-sumer. Examples include a predator-prey system in which the prey is able to kill or consume predator eggs or larvae and the insect pollinator and the host plant relationship in which the plants provide food, seeds, nectar and other resources for the pollinators while the pollinators have both positive and negative effects on the plants. By carrying out local analy-sis and bifurcation analysis of the model, we discuss the stability of the positive equilibrium and show that under some conditions a non-trivial periodic solution through Hopf bifurcation appears when the maturation parameter passes through some critical values. ∗Research was partially supported by NSFC (No. 11471044 and No. 11371058). †Research was partially supported by the French Ministry of Foreign and European Affair