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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)

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