MATH 481, Project 5
The Role of the Predator in Diversity

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The Taylor series expansion of a function $f : R^n \to R^n$ (that is, a vector field) introduces the concept of the Jacobian matrix, $Df$. Consider the nonlinear system of differential equations $dx/dt = f(x)$ associated with that vector field. Let $\bar{x}$ be an equilibrium. The Hartman-Grobman theorem asserts that the structure of the orbits in a sufficiently small neighborhood of $\bar{x}$ is homeomorphic to that of the linear system $d\xi/dt = A \xi$ near the origin, where $A = Df\big|_{\bar{x}}$, provided that no eigenvalues of $A$ lie on the imaginary axis in the complex plane.

Competing species

Explain how the competing species model \begin{align*} \frac{dx}{dt} &= (a_1 - b_{11} x - b_{12} y)x, \\ \frac{dy}{dt} &= (a_2 - b_{21} x - b_{22} y)y \end{align*} comes about and show that its behavior falls into one of four distinct cases, depending on the relative values of the coefficients.

Task 1: Plot a representative phase portrait for, and identify the equilibria of, each of the four cases.

Stability in mathematics versus stability in biology

The words stable and unstable are used with entirely different meanings in mathematics and mathematical biology. You will have to infer their meanings from their contexts.

Usage in mathematics

We know that a nonlinear differential equation can have many equilibria, some of which may be stable and some unstable. It should be clear that stability is an attribute of an equilibrium point, not that of a differential equation as a whole. You may say “this equilibrium is stable” or “that equilibrium is unstable”, but there is no meaning to the phrase: “this differential equation is stable”.

Usage in mathematical biology

In other disciplines the word “stable” may have other meanings. In mathematical biology, for instance, a multi-species population is said to be “stable” if the species can coexist forever, and “unstable” if one or more of the species are driven to extinction. In that sense, the system of differential equations that models the populations is said to be “stable” or “unstable”.

Gause's Competitive Exclusion Principle, formulated by the Russian biologist Gause in the 1930s based on experimental observations, states that a population of strongly competing species cannot be stable—one species will dominate and the others will become extinct. The adjective “strongly” requires some elaboration. I will do that in class.

Task 2: In the four distinct cases of the competing species noted above, determine which is stable and which unstable according to the mathematical biologist. The stability of the individual equilibria may be determined through linearization but that would be an overkill; for the purposes of this project it would suffice if you determine stability just by looking at the phase portraits.

Paradox of the plankton

There are well-known situations in nature that appear to conflict with Gause's Competitive Exclusion Principle, the most prominent of which is what became known in the 1960s as the Paradox of the plankton.

It was suggested that the paradox may be resolved by accounting for the the presence of a predator that preys simultaneously on the competing species. This led to:

Question: Is it possible to stabilize a biologically unstable population of two competing species by introducing a predator?

Enter the predator

Task 3: Let's limit the rest of the project to the case where $a_1/b_{11} < a_2 / b_{21}$ and $a_1/b_{12} < a_2 / b_{22}$.

Identify the three equilibria in the first quadrant. Apply the linearization analysis (the Hartman-Grobman Theorem) to determine the stability of those equilibria. Conclude that the population is unstable (in the sense of the biologist).

To investigate the effect of a predator on the population, Parrish and Saila (see the bibliography section below) introduced a predator, $z$, that preys on both $x$ and $y$, leading to the model \begin{align*} \frac{dx}{dt} &= ( a_1 - b_{11} x - b_{12} y - b_{13} z)x, \\ \frac{dy}{dt} &= ( a_2 - b_{21} x - b_{22} y - b_{23} z)y, \\ \frac{dz}{dt} &= (-a_3 + b_{31} x + b_{32} y - b_{33} z)z. \end{align*} Assuming $b_{13} = b_{23}$, $b_{31} = b_{32}$, and $b_{33}=0$ for simplicity, they performed several computer experiments toward answering the question posed above, but their investigation was inconclusive.

Two years later Cramer and May analyzed the same model and answered the question in the positive through a choice of a rather odd set of coefficients: \[ a = \begin{bmatrix} 3.0 \\ 2.1 \\ 1.2 \end{bmatrix}, \quad b = \begin{bmatrix} 9\times 10^{-5} & 3.0\times10^{-5} & 0.15 \\ 3\times 10^{-5} & 0.6\times10^{-5} & 0.15 \\ 0.6\times 10^{-4} & 0.6\times10^{-4} & 0. \end{bmatrix}. \]

Their article does not say how they arrived at those coefficients but from the looks of it, I conjecture that it was through clever guessing and trial and error.

One of the goals of this project is to produce mathematical techniques whereby we may find much neater coefficient values that we select from a prescribed set of integers. For instance, we find that the coefficients \[ a = \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix}, \quad b = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 1 & 1 \\ 3 & 3 & 0 \end{bmatrix}, \]

selected from the set $\{1,2,3\}$ satisfy the requirements and are much neater than Cramer and May's haphazard numbers. The key idea that helps to find such nice coefficients is the application of the Routh–Hurwitz stability criterion which is missing in the previous literature.

Task 4: Conduct a search for suitable coefficients from a four-tuple of integers assigned to you in the following table:

After selecting the coefficients, do:

1. Let $x(0)=1$, $y(0)=1$, $z(0)=0$ (predator absent). Plot $x(t)$ and $y(t)$ versus time and verify that one of the two species dies out.
2. Let $x(0)=1$, $y(0)=1$, $z(0)=1$ (predator present). Plot $x(t)$, $y(t)$, and $z(t)$ versus time and verify that all three species survive.

A 3D phase portrait

It is possible to produce a three-dimensional phase portrait with the help of Maple's DEplot3d(). I have done it but it's quite a bit of work. Do it if you feel like taking up the challenge. The blue dot in the diagram marks the stable equilibrium $(\bar{x},\bar{y},\bar{z})$ in the interior of the positive octant.

Bibliography

The links embedded in the bibliography items below lead to the PDFs of the corresponding articles. You may read them online or download them to your personal computer but probably you don't want to save them to your UMBC disk since these files are large and you may run out of disk space.

You may copy and paste these entries to your article's bibliography if you want, and you may add other references as you see fit.

@article{hutchinson-1961,
  author = {Hutchinson, G. E.},
  title  = {The Paradox of the Plankton},
  journal= {The American Naturalist},
  volume = {95},
  number = {882},
  date   = {1961},
  pages  = {137--145},
}

@article{paine-1966,
  author = {Paine, Robert T.},
  title  = {Food Web Complexity and Species Diversity},
  journal= {The American Naturalist},
  volume = {100},
  number = {910},
  date   = {1966},
  pages  = {65--75},
}

@article{parrish-saila-1970,
  author = {Parrish, J. D. and Saila, S. B.},
  title  = {Interspecific Competition, Predation and Species Diversity},
  journal= {Journal of Theoretical Biology},
  volume = {27},
  number = {2},
  date   = {1970},
  pages  = {207--220},
}

@article{cramer-may-1972,
  author = {Cramer, N. F. and May, R. M.},
  title  = {Interspecific Competition, Predation and Species Diversity: A~Comment},
  journal= {Journal of Theoretical Biology},
  volume = {34},
  number = {2},
  date   = {1972},
  pages  = {289--293},
}

@article{hofbauer-sigmund-1989,
  author = {Hofbauer, J. and Sigmund, K.},
  title  = {On the stabilizing effect of predators and competitors
            on ecological communities},
  journal= {Journal of Mathematical Biology},
  volume = {27},
  date   = {1989},
  pages  = {537--548},
}

Author: Rouben Rostamian