Disease Induced Oscillations between Two Competing Species

Abstract

Abstract. The interaction of disease and competition dynamics is investigated in a system of two competing species in which only one species is susceptible to disease. The model is kept as simple as possible, combining Lotka-Volterra competition between the species with disease dynamics of susceptible and infective individuals within one of the species. It is assumed that pure vertical disease transmission (from parent to offspring) dominates horizontal transmission (by contact between infective and susceptible individuals) and that infective individuals have the same competition strength as susceptibles but a lower intrinsic growth rate. These assumptions yield three-dimensional competitive Lotka-Volterra dynamics modeling the disease-competition interaction. It is proved that if in the absence of disease there is competitive exclusion between the two species, then the presence of disease can lead to stable or oscillatory coexistence of both species. The case of oscillatory coexistence can be viewed either as disease induced oscillations between competing species or as competition induced oscillations in an endemic disease. By contrast, conditions are found under which, if the two species coexist in the absence of disease, then the introduction of disease does not induce oscillations, and the long-term dynamics are determined by the basic reproduction number.

An immuno-eco-epidemiological model of competition

“…From η(τ ) = 0 it follows that n 1 = s 0 . The first equation in (20) changes to the following form:…”

“…The introduction of disease in the competition between two species has led to destabilization in the dynamics [20]. More recently, disease infecting a predator has been found to lead to complex dynamics and chaos [52].…”

Stability of the Endemic Coexistence Equilibrium for One Host and Two Parasites

“…Periodic solutions have even been found for two competing host species only one of which is afflicted by a parasite provided that the incidence is density-dependent [44,45]. One readily checks that the analogous model in [44] with frequency-dependent incidence has all trajectories converging towards an equilibrium. The mathematical mechanism for the relation between stability and frequency-dependent incidence in these papers is as follows: Rewriting the model in terms of the fraction of infectious (and possible removed) individuals, one can decouple a subsystem of at most two dimensions from the rest of the system and subject it to Poincaré/Bendixson and Dulac theory.…”

“…Similarly, undamped oscillations can occur in SIS endemic models with two hosts and one parasite for density-dependent, but not frequencydependent incidence [20,24,25]. Periodic solutions have even been found for two competing host species only one of which is afflicted by a parasite provided that the incidence is density-dependent [44,45]. One readily checks that the analogous model in [44] with frequency-dependent incidence has all trajectories converging towards an equilibrium.…”

“…One class of mechanisms is the presence of multiple hosts and/or multiple parasites or multiple strains of one parasite [3,4,20,24,26,27,28,44,45]. This paper continues the investigation of an endemic model with one host and two strains of one parasite (or two different parasites) where one strain is exclusively vertically transmitted (the VT strain) and the other strain is horizontally (and possibly also vertically) transmitted (the HT strain) [9,13].…”