Uni-directional consumer-resource theory characterizing transitions of interaction outcomes

A resource is considered here to be a biotic population that helps to maintain the population growth of its consumers, whereas a consumer utilizes a resource and in turn decreases its growth rate. Bi-directional consumer–resource (C–R) interactions have been the object of recent theory. In these interactions, each species acts, in some respects, as both a consumer and a resource of the other, which is the basis of many mutualisms. In uni-directional C–R interactions between two species, one acts as a consumer and the other as a material and/or energy resource, while neither acts as both. The relationship between insect pollinator/seed parasites and the host plant is an example of the latter interaction type of C–R, as the insect provides no material resource to the plant (though it provides a pollination service). In this paper we consider a different variation of the uni-directional C–R interaction, in which the resource species has both positive and negative effects on the consumer species, while the consumer has only a negative effect on the resource. A predator–prey system in which the prey is able to kill or consume predator eggs or larvae is an example. Our aim is to demonstrate mechanisms by which interaction outcomes of this system vary with different conditions, and thus to extend the uni-directional C–R theory established by Holland and DeAngelis (2009). By the analysis of a specific two-species system, it is shown that there is no periodic solution of the system, and the parameter (factor) space can be divided into six regions, which correspond to predation/parasitism, amensalism, and competition. The interaction outcomes of the system transition smoothly when the parameters are changed continuously in the six regions and/or initial densities of the species vary in a smooth fashion. Varying a pair of parameters can also result in the transitions. The analysis leads to both conditions under which the species approach their maximal densities, and explanations for phenomena in experiments by Urabe and Sterner (1996).