The more closely we look at the world the more clearly we see its interconnected nature. We can use the tools we’ve developed in this section to model interactions between distinct populations. We can model symbiotic interactions, such as the interaction between sea anemones and clown fish, or competitive interactions, such as the interaction between lion and hyena populations competing for small prey. We can model predator-prey interactions, such as that between hyenas and the Thomson’s gazelle, lions and water buffalo, or cats and mice. In the examples in this section we’ll simplify our models to focus on just two populations.
EXAMPLE 31.29
Consider the predator-prey relationship between hyenas and the Thomson’s gazelle, a small gazelle native to Africa. Let’s make the following simplifying assumptions.i. Assume hyenas are the gazelles’ major predator and that in the hyenas’ absence the gazelle population would grow exponentially.
ii. Assume gazelles are the major food source for hyenas; with no gazelles the hyenas would die off.
Model this interaction with a system of differential equations.
SOLUTION Let h(t) = the number of hundreds of hyenas at time t.
Let g(t) = the number of hundreds of gazelles at time t.
We can model the interaction by a system of differential equations of the form dh
dt = −k1h + k2hg dg
dt = k3g − k4hg
where k1, k2, k3, and k4are positive constants. Just as the rate of transmission of disease
is proportional to interactions between the infected and the susceptible, which is in turn proportional to the product of their numbers, so too is the rate of nourishing/fatal interaction between hyena and gazelle proportional to the product of their population sizes. Observe that if h = 0 thendgdt = k3gand if g = 0 then
dh
dt = −k1h. For the sake of concreteness, we’ll
work with the values of k1, k2, k3, and k4given below and analyze solutions in the gh-plane.
dh
dt = −0.3h + 0.1gh = 0.1h(−3 + g) dg
dt = +0.4g − 0.4gh = 0.4g(1 − h)
Nullclines: dhdt = 0 when h = 0 or g = 3. In the gh-plane this is where trajectories have horizontal tangent lines.
dg
dt = 0 when g = 0 or h = 1. In the gh-plane this is where trajectories have vertical
tangent lines.
Equilibrium points: dhdt = 0 at h = 0 or g = 3, so at an equilibrium point either h = 0 or g = 3.
31.5 Systems of Differential Equations 1039
Suppose h = 0. Then in order for dgdt to be zero we must have g = 0. (0, 0) is an
equilibrium point.
Suppose g = 3. Then in order for dgdt to be zero we must have h = 1. The point (3, 1)
is an equilibrium point.
Use the differential equations to orient the vertical and horizontal tangents. For instance, when g = 3 we know thatdgdt >0 for 0 < h < 1 and
dg dt <0 for h > 1. h g 1 3 Figure 31.30
The h and g axes are oriented as shown in Figure 31.30. The nullclines divide the relevant first quadrant region into four subregions. The direction of trajectories in each region is indicated in Figure 31.30.
From Figure 31.30 we see that for g(0) > 0 and h(0) > 0 the trajectories either spiral in toward (3, 1) or spiral outward, or are closed curves. From the slope field it appears that the curves are closed. Looking atdhdg =0.1h(−3+0.1g)0.4g(1−h) enables us to distinguish between these options. Separating variables, we find that
ln(h) − h = −3 ln(g) + g + C.
Suppose we start at the point (5, 1). We can solve for C and show that the trajectory through (5, 1) intersects the line g = 3 exactly twice, once for h > 1 and once for h < 1. Similarly we can show that it intersects the line h = 1 exactly twice, once for g < 3 and once for g >3. The trajectory through (5, 1) is a closed curve. In fact, all the trajectories in the first quadrant are closed.
g h
3 1
Figure 31.31
Our model predicts that the hyena and gazelle populations will oscillate cyclically. When there are few gazelle the hyena population decreases due to lack of food. The decrease
in the number of hyenas allows the gazelle population to thrive, but as the gazelle population increases the hyenas’ food source is replenished, allowing the hyena population to flourish. This flourishing takes its toll on the gazelles, and the cycle repeats.
To the extent that a model reflects observed population dynamics, it is a good model. Models that don’t reflect observed behavior must be modified. There are various ways this predator-prey model can be modified. For instance, there may be competition among gazelle for limited grazing land so that in the absence of hyena the gazelle population exhibits logistic growth. This can be reflected in the system of differential equations by inserting a −k5g2term as shown.
dh
dt = −k1h + k2hg dg
dt = k3g − k4hg − k5g2,
where the −k5g2term (k5>0) reflects competition among gazelles.
The predator-prey system of differential equations given in Example 31.29 is sometimes referred to as Volterra’s model after the Italian mathematician Vito Volterra (1860-1940) who was encouraged to analyze the predator-prey relationship between sharks and the fish they prey upon by his son-in-law, the biologist Humberto D’Ancona.10By inserting a term in each equation to account for fishing, Volterra was able to explain why fishing conducted in the Adriatic Sea was raising the average number of prey and lowering the average number of predators over any cycle. Similar analysis has been successfully used to analyze unexpected results of introducing DDT into a predator-prey system, leaving predator populations lowered and prey populations elevated.
Competition between species can be modeled by differential equations of the form dx dt = k1x − k2xy dy dt = k4y − k5xy or dx dt = k1x − k2xy − k3x2 dy dt = k4y − k5xy − k6y2
where k1, k2, . . . , k6are positive constants. The latter set of differential equations takes
into account competition between members of the same species in addition to competition between species.