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1. Qualitatively analyzing a one-dimensional model (2 marks)

Consider the differential equation $$\frac{dx}{dt} = x(1-x)$$.

1. Use R to plot $$\frac{dx}{dt}$$ vs. $$x$$ for $$-1 \le x \le 2$$. (0.5 marks)

2. Sketch a one-dimensional phase portrait on paper. Mark the direction of the derivative in each region with arrows and label the fixed points as “stable” or “unstable”. (0.5 marks)

3. Qualitatively graph $$x$$ vs. $$t$$ for the following three initial conditions: $$x_0 = -0.5$$, $$x_0 = 0.5$$, and $$x_0 = 1.5$$. What happens as $$t$$ becomes large in each case? (1 mark)

1. The Allee Effect (3 marks)

Generally, as population size increases, a population will experience a decreased growth rate due to greater competition for resources. This is a negative density-dependent growth rate, and one example of this is the logistic model.

The Allee effect introduces positive density dependence, where increases in population size result in increased growth rates over a certain range of population sizes. The Allee effect can be incorporated into the logistic growth equation as follows:

$\frac{dN}{dt} = rN\left(1-\frac{N}{K}\right)\left(\frac{N-A}{K}\right)$

Here $$r$$ represents the growth rate of the population, $$K$$ is the carrying capacity, and $$A$$ is the critical size above which the total growth rate is positive.

1. Take $$r=1$$, $$A=10$$, and $$K=50$$. Use R to plot $$\frac{dN}{dt}$$ vs. $$N$$ for $$0 \le N \le 55$$. For which values of $$N$$ is the growth rate ($$\frac{dN}{dt}$$) positive or negative? (0.5 marks)

2. Sketch a one-dimensional phase portrait on paper, leaving $$A$$ and $$K$$ as variables (i.e. without choosing values for $$A$$ and $$K$$, but with $$0 \le A < K$$). Mark the direction of the derivative in each region with arrows and label the fixed points as “stable” or “unstable”. (1 mark)

3. Use R to plot the per capita growth rate ($$\frac{1}{N}\frac{dN}{dt}$$) vs. $$N$$ for this model of the Allee effect and for the logistic growth model: $$\frac{dN}{dt} = rN(1-\frac{N}{K})$$.

• What do you notice about the density ($$N$$) dependence of the per capita growth rate in each case? Hint: in the logistic model, the growth rate per capita (per organism) decreases in a straight line as $$N$$ increases.

• What happens to the Allee effect as $$A$$ decreases? Plot curves for $$A=0$$ and a few values of $$A>0$$.

The parameter $$A$$ controls the strength of the Allee effect: for $$A > 0$$, the Allee effect is said to be strong, and for $$A=0$$, the Allee effect is weak.

• What will happen to a population experiencing a strong Allee effect if the population size falls below $$A$$? What will happen to a population experience a weak Allee effect if the population size falls below $$A$$? (1 mark)

4. Describe two biological situations in which you might expect to see an Allee effect (either weak or strong). (0.5 marks)

1. Analyzing a two-dimensional model (3 marks)

Consider the Lotka-Volterra predator-prey model:

$\frac{dx}{dt} = x - \alpha y x$

$\frac{dy}{dt} = \beta y x - \delta y$

where $$x$$ and $$y$$ are the prey and predator population sizes, respectively. $$\alpha$$ is the rate at which prey are killed by predators, $$\beta$$ is the rate at which predators grow by consuming prey, and $$\delta$$ is the death rate of predators.

1. Numerically solve this system using the ode function from the deSolve package. Use the parameters $$\alpha = 0.6$$, $$\beta = 0.9$$, and $$\delta = 0.5$$. Choose $$x_0 = y_0 = 0.5$$ as the inital state for the system. Choose a range for time that shows the behaviour of the system. On the same axes, plot $$x$$ vs. $$t$$ and $$y$$ vs. $$t$$. Label $$x$$ and $$y$$ and the axes. (1 mark)

2. Numerically solve the system again, but this time use a very specific starting point: choose $$x_0 = \delta / \beta$$, and $$y_0 = 1/\alpha$$.

• Again plot your result: $$x(t)$$ and $$y(t)$$ on the same axes. What’s different? (1 mark)
• The point $$x_0 = \delta / \beta$$, $$y_0 = 1/\alpha$$ is a fixed point. Is it stable, unstable, or something else altogether? Justify your answer by referring to the two plots you made. (1 mark)