title: On the stability of polymorphic host-pathogen populations.
abstract: The stability of populations of hosts and micro-parasites is investigated where each consists of n varieties that are equal in every respect except that each strain of parasites can infect only one specific strain of hosts and none of the others. Collectively the host strains are limited by a carrying capacity and through this limitation the host populations interact with each other. Hosts are assumed to reproduce asexually or such that different strains do not mate or are not fertile if they do. When the excess death rate caused by the pathogenic parasites is sufficiently large, then the host population is regulated to an equilibrium below the carrying capacity of the environment. This polymorphic equilibrium is shown to be locally asymptotically stable. When one of the parasite strains is absent, then all the other strains die out asymptotically. However, if host resistance to all infectious strains of parasites is achieved at the cost of a lower birthrate of the resistant host strain, then, if a certain condition for the various parameters is satisfied, stable coexistence between infected and resistant hosts is possible. There are many examples where susceptibility and resistance of hosts depends upon the conformation of specific proteins that are involved in host-parasite interactions and hence upon alleles at genetic loci that code for these proteins. We propose that polymorphism in wildtype populations which has been the subject of much theorizing in mathematical genetics may be due to host-pathogen interactions. Our model suggests how a polymorphic population, once established, can remain polymorphic indefinitely.
mesh_terms: D000818:Animals; D004198:Disease Susceptibility; D006790:Host-Parasite Interactions; D008433:Mathematics; D008954:Models, Biological; D011110:Polymorphism, Genetic