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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 each of the four cases. Mark the nullclines and equilibria within the first quadrant.
The words stable and unstable are used with entirely different meanings in mathematics and mathematical biology. You will have to infer their meanings from context.
Gause's Competitive Exclusion Principle, formulated by the Russian biologist Gause in the 1930s based on experimental observations, states that a population of competing species cannot be stable—one species will dominate and the others will die out.
Task 2: In the four distinct cases of the competing species noted in Task 1, determine which is stable and which is unstable according to the mathematical biologist. The stability of the individual equilibria may be determined through linearization as usual 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.
Remark: You will find that the case “$a_1/b_{11} < a_2/b_{21}$ and $a_2/b_{22} < a_1/b_{12}$” of Task 1's four cases apparently violates Gause's principle as it exhibits a stable equilibrium where the competing species do coexist. I will explain in class why this case falls outside the scope of Gause's principle and therefore is not a violation.
Although Gause's Competitive Exclusion Principle seems to be valid in laboratory setting, it appears to conflict with many observed situations in nature, the most prominent of which is what became known in the 1960s as the paradox of the plankton. (Read about it in Wikipedia.)
It was hypothesized by Hutchinson in 1961 and Paine in 1966 (among others) 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 the unstable population of two competing species by introducing a predator?
An attempt to answer that question was made by Parrish and Saila in 1970 albeit their results were inconclusive. A successful attempt was made by Cramer and May in 1972. In this project we will fine-tune Cramer and May's result through a more detailed mathematical analysis.
Following Parrish and Saila, and Cramer and May, we focus on the case “$a_1/b_{11} < a_2 / b_{21}$ and $a_1/b_{12} < a_2 / b_{22}$” which, as you must have discovered by now, corresponds to an unstable (in the sense of the biologist) population.
To investigate the possibility of stabilizing that population, Parrish and Saila 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 called this the equal predation assumption—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 assume that it was through clever guessing and trial and error.
The major goal of this project is to 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. Compare that to Cramer and May's haphazard numbers. The key idea that helps us find such nice coefficients is the application of the Routh–Hurwitz stability criterion which is missing in the previous literature.
Task 3: What are the necessary and sufficient conditions on the coefficients of the cubic polynomial $ \lambda^3 + c_1 \lambda^2 + c_2 \lambda + c_3 $ for its roots to lie on the left-hand side of the complex plane?
Task 4: Conduct a search for suitable coefficients from a four-tuple of integers assigned to you in the following table:
| Adi {1, 2, 4, 5} |
Andrew {1, 2, 4, 6} |
Connor {1, 2, 4, 7} |
Emma {1, 2, 5, 6} |
Fadlulah {1, 2, 5, 7} |
| Gabriel {1, 2, 6, 7} |
Ignatius {1, 3, 4, 5} |
Julian {1, 3, 4, 6} |
Katie {1, 3, 4, 7} |
Megan {1, 3, 5, 6} |
| Nate {1, 3, 5, 7} |
Nick {1, 3, 6, 7} |
Nishwanth {1, 4, 5, 6} |
Owen {1, 4, 5, 7} |
Peyton {1, 4, 6, 7} |
| Rijen {1, 5, 6, 7} |
Theo {2, 3, 4, 5} |
Xavier {2, 3, 4, 6} |
After selecting the coefficients, do:
Task 5 (optional): On page 290 of Cramer and May's article they state (here I have translated their notation into ours):
In the absence of predation it is well known that the two-species system is unstable with regard to competition if \[ b_{11} b_{22} - b_{12} b_{21} \le 0 \] and stable otherwise.
Do you believe that “well known” fact? Try $a_1 = 6$, $a_2 = 1$, $b_{11} = 3$, $b_{12} = 2$, $b_{21} = 1$, $b_{22} = 1$.
How would you correct their statement? Hint: Your findings under Task 2 should help.
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.
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Here are BibTeX entries for the articles of Parrish and Saila, and Cramer and May. You may copy and paste these into your article if you want. Add other references as you see fit,
@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},
year = {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},
year = {1972},
pages = {289-293},
}
| Author: Rouben Rostamian |