An application of predatorprey models harvesting populations

Much of our understanding of the harvesting of wild populations has come from the fisheries literature, with classic long-term studies in the North Sea, Atlantic and Pacific. These studies have used both continuous- and discrete-time population models. These have been complemented by studies on terrestrial populations including high-profile animals such as the African elephant. In this section we will consider how population models can be used to determine appropriate levels of harvesting. We will start with an unstructured model in continuous time and then explore how much more can be learned from a structured model.

6.5.1 Unstructured population models

Assume that population growth is continuous and described by the logistic equation. Although in many cases the logistic equation is too simple a description of population change (see discussion in Chapter 5), it provides a useful entry point to understanding the dynamic possibilities of harvesting. We begin by plotting the rate of population growth, dN/dt, against population size, Nt (Fig. 6.13).

You should note that the curve in Fig. 6.13 is a parabola, which is the shape generated by a quadratic equation (recall that the right-hand side of the logistic equation in expanded form is rNt - rNt2/K). All population sizes which yield values of dN/dt > 0 can, in theory, be subject to harvesting. This means that a fraction of any growing population should be able to be harvested without causing the population to become extinct. The maximum sustainable yield occurs where the maximum value of dN/dt occurs; that is, when the population is growing most rapidly. This result is generally true for all population growth curves (see the discussion in May & Watts 1992).

Fig. 6.13 Population growth dN/dt described by the logistic equation plotted against population size (N). K is the carrying capacity.

To determine the population density corresponding to the maximum sustainable yield we note that the maximum occurs when the gradient of the curve equals 0:

We will drop the subscript t to make the equations easier to read (N will still mean the population density at time t). For the logistic curve in Fig. 6.13 the derivative of the expression rN - rN2/K with respect to N is r - 2rN/K. Therefore r (1 - 2( N/K )) = 0

The solution to this equation is either r = 0 (a trivial solution) or (1 -2(N/K)) = 0. Rearranging the latter yields the solution that N = K/2. So the population is growing at its fastest when it is half its maximum density. This agrees with the shape of the logistic curve where the point of inflexion (the maximum gradient) was at K/2 (Figs 5.11 and 6.13).

To find the maximum value of dN/dt we need to substitute the size at which the maximum occurs (K/2) for N in the logistic equation dN/ dt = rN(1 - N/K). This gives dN/dt = rK/4. This method of finding the maximum value is useful if the function for population growth is more complex than the logistic equation.

To develop these ideas let us assume a general form of population growth equation dN/dt = f(N) which describes the rate of change in population size as some function f) of population size. If this function f(N) has a maximum value or values of dN/dt over a certain range of values of N then differentiation can be used to find the maximum value. You should note that the criterion of d(dN/dt)dN = 0 (equation 6.16) is not sufficient to identify a maximum value. It might equally identify a minimum value or a point of inflexion. A harvesting function can be incorporated into the general population growth equation (Beddington 1979):

where h(N) gives the reduction in dN/dt at a particular value of N due to harvesting. The population change and harvesting functions of the generalized form of equation 6.17 can be combined on one graph as they are both functions of N (Figs 6.14 and 6.15). We will now consider several harvesting possibilities with a logistic growth curve.

If h(N) is constant and therefore independent of N, harvesting is represented by a horizontal line on the graph. We know that sustainable harvesting can only occur when dN/dt > 0; when f(N) > h(N) in equation 6.17. The area on the graph when this condition is met is shaded in Fig. 6.14 and lies between N = 20 and N = 80. What happens to the population at different population sizes? For example, consider a population of size 70. Here f(N) > h(N) and therefore population size increases (dN/dt > 0, equation 6.17).

Fig. 6.14 Constant harvesting h(N) and logistic population growth f(N) plotted against N (equation 6.17). The shaded area shows where f(N) > h(N). Where the two lines (functions) intersect the rate of population change is 0.
Fig. 6.15 Linear increase in harvesting (h(N)) and logistic population growth (f(N)) plotted against population size (N).

In other words, from population size 70 we move to the right on the graph. Conversely, for a population of size 90, f(N) < h(N) and so dN/dt < 0 and the population size decreases.

By this reasoning it can be seen where h(N) = f(N) at N = 80 (and therefore dN/dt = 0) there is a locally stable equilibrium point. A population which receives a small displacement away from that equilibrium point will tend to return to it. What then of the other point at which dN/dt = 0, at N = 20? If

N< 20 then f(N) is less than h(N) and therefore dN/dt < 0. So, the population will continue to decrease if reduced below 20 until it reaches N = 0, which is local extinction (N = 0 is effectively a third equilibrium point, which is locally stable). If the population size is increased above N = 20 then f(N) > h(N) and therefore the population size will continue to increase until it reaches the stable equilibrium point at N = 80.

Even the very simple scenario of constant harvesting combined with logistic growth provides the dynamic possibilities of extinction (below N = 20) and local stability (at N = 80). A second possibility for the harvesting function is that it increases linearly with prey population size (Fig. 6.15), so that the more fish there are, the more people go fishing. Again, N = 80 is seen as a stable equilibrium. In both this and the previous example, if the prey population is pushed beyond K (100) then it is predicted from the model that it will return towards K, but it will not stay there as h(N) is still greater than f(N) and therefore it continues to return to 80.

6.5.2 Structured population models

We begin by considering an unstructured population in discrete time in which the population dynamics are described by X, the finite rate of population change, which is equivalent to the dominant eigenvalue of the population projection matrix for a structured population. We know that a population with X < 1 will decline in numbers, so if a harvesting policy is to be sustainable it should not decrease the value of X below 1. It is therefore possible to arrive at a simple definition of the maximum amount of a population which can be harvested. For example, if X = 2, the maximum amount which can be removed is that which keeps X = 1:

If X = 3, the maximum fraction of the population which could be harvested is 2/3. This assumes that X is constant from year to year and that a constant fraction is removed in each time period.

Now consider the implications of age, size or stage structure. To explore the effect of population structure on harvesting consider the two-stage model of biennial plants used previously (equations 4.9 and 4.10):

The characteristic equation was X2 = fs01s12 (equation 4.17). We know that the maximum fraction which can be harvested is given by (X - 1)/X, so it becomes of interest to see how manipulation of f, s01 and s12 affects (X - 1)/X.

Assume that a fraction m1 of flowering plants is harvested prior to setting seed and therefore the fraction of surviving plants is represented by the fraction (1 - m1). Note that removal of flowering plants after seed set for a monocarpic species is not going to affect the population dynamics. This harvesting mortality can be incorporated into the model as follows:

vt+1 M with harvesting vt of F

In a similar way, we might imagine that a fraction m2 of rosette plants is harvested (of course, either m1 or m2 or both can have zero values). This can also be included in the model:

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