The Kaibab narrative: management on ill-understood systems

History of Kaibab plateau

Numerous models of ecosystems have been made – but sometimes one wonders whether somewhere a reality can be found which more or less is described by such a model. The term empirical validation may be too strong, the aim is model-based story-telling. There are some interesting real-world stories to tell about the ecological predator-prey dynamics. Perhaps, it supports the use of wise(r) or no interventions. Here are two such interesting stories.

Prior to 1907, the deer herd on the Kaibab plateau, which consists of some 727,000 acres and is on the north side of the Grand Canyon in Arizona, numbered about 4,000. In 1907, a bounty was placed on cougars, wolves and coyotes – all natural predators of deer. Within fifteen to twenty years, the larger part of these predators (over 8,000) was wiped out and a consequent and immediate irruption of the deer population followed (see Figure below). By 1918, the deer population had increased more than tenfold. The evident overbrowsing of the area brought the first of a series of warnings by competent investigators, none of which produced a much-needed quick change in either the bounty policy or the policy dealing with deer removal. In the absence of predation by its natural predators (such as cougars, wolves and coyotes) or by man as a hunter, the herd reached 100,000 in 1924; in the absence of sufficient food, 60 percent of the herd died off in two successive winters. By then, the girdling of so much of the vegetation through browsing precluded recovery of the food reserve to such an extent that subsequent die-off and reduced natality yielded a population about half that which could theoretically have previously maintained (Roberts et al. (1983); Meadows (1986)).

In the article The Rise and Fall of a Reindeer Herd, Scheffer (1951) tells a similar story about forty reindeer placed in 1911 on the two Pribiloff Islands, St. Paul and St. George, in Alaska to provide the native residents with a sustained source of fresh meat. There were no predators except humans. The herd on St. George grew quickly to 222 animals, after which it declined to a stable level of forty to fifty animals. The herd on the larger St. Paul increased to more than 2,000 animals by the late 1930s, after which it collapsed to only eight animals in 1950. It turned out that the reindeer survived in winter by eating certain shrublike lichen. When the herd size increased, the reindeer consumed the lichen faster than their regeneration rate and it needed only one severe winter to exterminate most animals on St. Paul.

Figure 1 The deer population on the Kaibab plateau population and the reindeer population on two Alaskan islands. 

Model: The Lotka-Volterra predator-prey equation

One of the early mathematical models to explain cycles in ecosystem populations was formulated by Lotka and Volterra in the 1920s. The model has actually been borrowed from chemical kinetics and provides in turn a metaphor for more complex processes, for example, in economic and social systems – remember the arms race model.The basic equations are:

 

(1a)

(1b)

 

The coefficients r, b, c and d make up the so-called ecological community matrix. They represent exponential growth with a net growth rate dependent on the population of the other species. By setting dX/dt = 0 and dY/dt = 0, one finds the attractors or equilibrium states, meaning the state in which both populations do not change. One stable state is the trivial (0,0) and the other is (c/d,r/b). When there is prey abundance, the predator population increases. The prey population starts declining due to predator abundance, after which predator population will go down again. The populations will thus oscillate over time, as is shown in Figure 1a-d.

Figure 1a,b (left). Simulated populations of prey (1) and predator (2) for the system description of equation 13.1 (a = 0,004; b = 0,16; d = 0,1; k = 0,3; Δt = 0.25, Runge-Kutta4). The populations oscillate permanently around the equilibrium values (3,4).

Figure 1c,d (right). Simulated populations of prey (1) and predator (2) for the system description of eqn. 13.1 (a = 0,004; b = 0,16; d = 0,1; k = 0,3; Δt = 0.25, Runge-Kutta4). The populations oscillate permanently around the equilibrium values (3,4) but switch to a new trajectory after a permanent disturbance (harvesting 3 prey, that is, 3 percent) is introduced in quarter 180.

These oscillations are rather ephemeral because the actual path is sensitive to the initial populations, as is seen in the phase diagram with trajectories for three different initial predator populations (Figure 1a-b). The system is neutrally stable: a delicate balance between stability and instability. If prey is harvested, the oscillations are dampened or even disappear altogether. Invasion of prey causes the oscillations to be amplified (Figure 1c-d). Invasion of predators also stabilises the system, but harvesting them tends to destabilise it. In other words, population sizes easily crash or explode under small perturbations. In other words: it is an extreme simplification.

Model simulations like this show that mathematical on the one hand can explain certain system behaviour, but are on the other hand such rigorous simplifications that hardly any conclusion about real-world behaviour can be drawn.

 

Literature

Case, T. (2000). An Illustrated Guide to Theoretical Ecology. Oxford, Oxford University Press

Edelstein-Keshet, L. (1988). Mathematical models in biology. New York, Random House.

Meadows, D. (1986). Personal communication.

Scheffer, V. (1951). The Rise and Fall of a Reindeer Herd. The Scientific Monthly 73(6)356-362