Population Dynamics of

2 Preys 1 Predator Models


In the process of modeling a food net of one predator, two preys, the behavior of the predator must be carefully considered.  For example, the predator may simply hunt for whatever prey it comes to first, or it may be restricted in what prey it can hunt for by the time of day, month, or year, or other environmental factors.  These behavioral differences lead to distinct predation considerations in mathematical models, leading to vastly different dynamical phenomena of the system.  The predation term models the rate at which the predator consumes the prey according to Holling’s theory on species predation.


These models can be analyzed using techniques such as singular perturbation analysis, and numerical computation.  Our study suggests that the ‘mixed’ predation model, having only one time unbiased predation term, has relatively simple behavior and that it may be possible to group the two preys together.  This would reduce the system to a traditional two-dimensional, one predator, one prey system.


The other model, where the predator must schedule its hunt between two preys leads to two ‘independent’ predation terms. Its behavior is much more complex.  It can have equilibrium point, limit cycles of distinct types, and chaotic dynamics.  Specifically, this model has a chaotic attractor containing a Shilnikov’s orbit as shown.


More work is needed on the independent predation model to determine if other types of chaotic behaviors are possible and their ecological implications.



Collaborators: Brian Bockelman, Elizabeth Green, Leslie Lippitt, and Jason Sherman, Bo Deng, and Wendy Hines


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Dynamics of Competition, Predation, Prolificacy of Food Webs


In this project we set out to understand the dynamical forms of competition between two predators, Y, Z, for a common prey, X, for which one of the predators, Y,  is the prey of a top-predator, W.  The XYW interaction forms a food chain, whereas the

XYZ forms a food web. 


A mathematical model is constructed for the study. It is based on two fundamental modeling principles in ecology. The prey is modeled according to the logistic growth model. All the predators are modeled according to the Holling Type II predation functional form.


Complex as it is, qualitatively equivalent dynamics can be cataloged according to 3 characteristics of the species: competitiveness, predatory efficiency, and reproductive strength. As examples to illustrate these concepts, we say the predator Z is competitive if all the XY-attractors are unstable in the expanded XYZ-web. We say the predator Y is efficient if the XY-attractor is a stable cycle, and weak if the XY-attractor is a stable equilibrium point. To measure the reproductive strength, we use the ratio of the maximum growth rate of Y to the maximum growth rate of X and call it the YX-prolificacy parameter. 

Important findings are as follows: If both Y and Z are weak, the classical Competition Exclusion Principle implies that Y, Z cannot be both competitive and one must die out. In the case that Z dies out in the XYZ-web, we demonstrated that it can invade an XYW-chain where W is a top-predator of Y. The coexistence states in the XYZW-web can be steady state, cycle, and chaos. In particular, as the ZY-prolificacy parameter increases, the coexisting state bifurcate from equilibrium points to periodic cycles to chaotic attractors. Two types of coexisting chaotic attractors are shown. The one on the right occurs when W is efficient. The one on the left occurs when W is weak.  In other words, chaos arises from a deceptively simple arrangement in the latter case: without Z the XYW dynamics is an equilibrium, and without W the XYZ dynamics reduces to an XY-equilibrium state with Z going extinct. Yet, by out reproducing its competitor Y, Z can invade the chain XYW, and establish a chaotic coexistence for the expanded web.


Numerous numerical experiments were carried out. More importantly rigorous proofs were also obtained for the existence of equilibrium, cycle, and chaotic states. 


Collaborators: Brian Bockelman, Elizabeth Green, Leslie Lippitt, and Jason Sherman, Bo Deng, and Wendy Hines  


(Download Preprint)