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Turing Instability and Colony Formation in Spatially Extended Rosenzweig–MacArthur Predator–Prey Models with Allochthonous Resources

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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 paradox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations.

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Authors and Affiliations

  1. Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA

    Zhi Zhou

  2. Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9054, New Zealand

    Robert A. Van Gorder

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  1. Zhi Zhou
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Correspondence to Robert A. Van Gorder.

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Zhou, Z., Van Gorder, R.A. Turing Instability and Colony Formation in Spatially Extended Rosenzweig–MacArthur Predator–Prey Models with Allochthonous Resources. Bull Math Biol 81, 5009–5053 (2019). https://doi.org/10.1007/s11538-019-00667-0

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