Date: Aug. 23, 2018
Allochthonous resources and mathematical modeling of population dynamics
Allochthonous resources refer to resources imported to an ecological system and they appear in many ecological food webs. The most typical example of allochthonous resources is the population dynamics of arctic foxes, lemmings, and seal carcasses on the coastal habitats. Ecologists believe that arctic foxes subsist on both lemmings and seal carcasses discarded by polar bears. The seal carcasses serves as allochthonous resources (food subsidies) in the typical predator-prey ecosystem containing arctic foxes and lemmings.
To better understand and predict the behaviors of interacting species, biologists and ecologists turn to mathematical modeling. Many mathematical models of the population dynamics in ecosystems have been proposed and thoroughly investigated. In this work, we develop a spatially extended population model to study the population evolution and distribution in the predator-prey-subsidy ecosystems. Specifically, we examine the behavior of colony formation in the predator and prey populations when allochthonous resources are present.
Turinmg and tURING-HOPF PATTERNS
Turing instabilities occur when a steady-state, stable in the absence of diffusion, become unstable when diffusion is present. Turing instabilities can induce patterns in biological systems, which are solutions that are temporally stable and spatially heterogeneous. When a Turing instability occurs near a Hopf bifurcation (the birth or death of a periodic solution or a limit cycle), complicated spatiotemporal patterns may emerge as a result of the nonlinear interaction of the two instabilities. This research investigates the conditions permitting Turing and Turing-Hopf patterns in the spatially extended predator-prey-subsidy models and numerically simulates the emergent patterns.
Abstract
While it is somewhat well known that spatial PDE extensions of the Rosenzweig– MacArthur predator–prey model do not admit spatial pattern formation through the Turing mechanism, in this paper we demonstrate that the addition of allochthonous resources into the system can result in spatial patterning and colony formation. We study pattern formation, through Turing and Turing–Hopf mechanisms, in two distinct spatial Rosenzweig–MacArthur models generalized to include allochthonous resources. Both models have previously been shown to admit heterogeneous spatial solutions when prey and allochthonous resources are confined to different regions of the domain, with the predator able to move between the regions. However, pattern formation in such cases is not due to the Turing mechanism, but rather due to the spatial separation between the two resources for the predator. On the other hand, for a variety of applications, a predator can forage over a region where more than one food source is present, and this is the case we study in the present paper. We first consider a three PDE model, consisting of equations for each of a predator, a prey, and an allochthonous resource or subsidy, with all three present over the spatial domain. The second model we consider arises in the study of two independent predator–prey systems in which a portion of the prey in the first system becomes an allochthonous resource for the second system; this is referred to as a predator–prey–quarry–resource–scavenger model. We show that there exist parameter regimes for which these systems admit Turing and Turing–Hopf bifurcations, again resulting in spatial or spatiotemporal patterning and hence colony formation. This is interesting from a modeling standpoint, as the standard spatially extended Rosenzweig–MacArthur predator–prey equations do not permit the Turing instability, and hence, the inclusion of allochthonous resources is one route to realizing colony formation under Rosenzweig–MacArthur kinetics. Concerning the ecological application, we find that spatial patterning occurs when the predator is far more mobile than the prey (reflected in the relative difference between their diffusion parameters), with the prey forming colonies and the predators more uniformly dispersed throughout the domain. We discuss how this spatially heterogeneous patterning, particularly of prey populations, may constitute one way in which the para- dox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations.
Keywords
Colony formation, Turing instability, Turing–Hopf instability, Rosenzweig–MacArthur model, Allochthonous resources