Figure 6.12: Phase-plane diagram showing a typical trajectory for the predator-prey problem enclosing the equilibrium point.
value. Thus the predator oscillation cycle always lags behind the prey oscillation, no matter what parameter values or initial conditions we use.
A feature of this model is that if you suddenly perturb the populations, they begin to move on a different phase-plane trajectory. This is one undesirable feature of the Lotka– Volterra equations from a biological perspective. Biologists would prefer a more realistic model where the populations tend to return to the original phase-plane curve after they have been perturbed slightly. This can be achieved using a modified version of the Lotka–Volterra equations, which we consider in Chapter 8.
Summary of skills developed here:
• Use the chain rule to prove that the trajectories of the the Lotka–Volterra equa- tions are always closed.
• Consider the points of intersection between two simple graphs to infer infor- mation about solutions to a more complicated equation.
6.7
Case Study: Bacteria battle in the gut
Simple models can often be applied to complex systems and successfully predict phenomena that have been observed. In the following case study we model, using the mechanisms discussed in this book, the interaction between different strains of Escherichia coli in the gut of animals. With this simple approach we show that a gut with a slow turnover rate favours different strains from one with a fast turnover rate, where these rates of turnover relate to diet. Adapted from Barnes et al. (2007).
Colicins are a class of protein antibiotics known as bacteriocins that are produced by certain strains of Escherichia coli (E. coli). They are produced through a process called cell lysis, where a single cell of bacteria produces many colicins. The advantage of producing colicins is to kill off other competing strains of E. coli in the same environment, although in doing so the colicin itself is destroyed. Thus colicins are important in mediating intra- specific interactions.
E. coli are found in the gut of animals, and colicin production varies markedly between populations. The process of cell lysis has been studied in detail by Levin (1988), Frank (1994), Gordon and Riley (1999), Kerr et al. (2002) and Kirkup and Riley (2004); how- ever, in this case study we investigate a different phenomenon. We examine whether the turnover rate in the gastro-intestinal tract determines which strains of E. coli dominate when colicin producing strains interact with colicin sensitive strains. It has been observed that in hosts with fast gut turnover rates, such as carnivores, non-colicin producing strains dominate, while in hosts with slow turnover rates, such as herbivores, colicin producing strains dominate.
To examine the dynamics, we consider a gastro-intestinal tract of fixed volume V and consider the interaction between two strains of E. coli: one a colicin producing strain x with density X in the gut, and one a colicin sensitive strain y with density Y . We assume a constant flow rate of food F into and from the gut, and the densities with which the strains enter with the flow are Xinand Yin. Within the gut we assume exponential growth for each
strain with growth rates of βx and βy, which is a reasonable assumption during the initial
stages of the dynamics. It should be noted that the qualitative results were unchanged when logistic growth was considered (see Exercise 6.22).
We now introduce the process of cell lysis. We assume that cells of strain x lyse at the per-capita rate αx, with each lysed cell producing 106colicin molecules, and each molecule
capable of killing a cell of the opposing strain, destroying itself in the process. A model for this process is dX dt = βxX + (Xin− X) F V − αxX, dY dt = βyY + (Yin− Y ) F V − c2XY. (6.24)
Note that the last term of dY /dt, c2XY is a mortality rate of Y cells due to the rate
of contact between the two different strains of cells. From Gordon and Riley (1999), the probability of an encounter of a colicin cell with a sensitive cell is of the order 10−11. We
can therefore determine the form of the c2XY terms and estimate the c2 parameter. We
can calculate the rate of of Y cells dying from a colicin molecule produced by all the lysed X cells as ( rate of Y cells dying ) = ( rate of X cells lysed ) × ( no. of colcin molecules produced per X cell )
×nprobability ofcontact o×nY cellsno. ofo ≃ (αxX) × 106× 10−11× Y
= 10−5α XXY.
Since typical cell densities are of the order 106, we rescale both X and Y in the equations
(6.24) to be in units of 106cells. This now gives c
2≃ 10αx(i.e., a factor of 106cancels out
from all terms on both sides of each equation except for the XY term).
Our aim is to investigate whether the mean residence time τ = V /F (where turnover rate is F/V = 1/τ ) determines which strain dominates. To do so we examine the nullclines, by setting dX/dt = 0 and dY /dt = 0, and consider any stable equilibrium points in the positive phase-plane. Our results are illustrated in Figure 6.13, where the stable node is plotted against F (recalling that F/V is turnover rate that increases with F ). We note that the ‘change’ between the dominance of colicin producing cells (X) and non-colicin producing cells (Y ), according to this model, occurs for F ≈ 6. It is evident that for low turnover rates (herbivores), the colicin producing strain dominates, while for high turnover rates (carnivores), the non-colicin producing strain dominates. This is in agreement with
6.8 Exercises for Chapter 6 171