The Predator-Prey Cycle: A Case of Overabundance
A wildlife reserve has a population of 200 deer and 20 wolves. The deer population grows at a rate of 0.5 per year (exponential) in the absence of wolves. Wolves prey on deer, and each wolf consumes on average 10 deer per year. The wolf population grows at a rate of 0.1 per year per deer (i.e., growth rate is proportional to deer abundance). The reserve has a carrying capacity of 500 deer due to limited food. A new policy proposes to cull 50 deer per year to reduce overgrazing. The manager asks: (1) What is the current growth rate of the deer population? (2) How will the culling affect the deer and wolf populations over time?
1. Calculate the current growth rate of the deer population, considering both wolf predation and the carrying capacity. Show your reasoning.
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The deer population growth can be modeled with a logistic growth equation including predation: dD/dt = rD(1 - D/K) - pDW, where r=0.5, K=500, D=200, W=20, and p is the per capita predation rate. Each wolf consumes 10 deer per year, so total predation = 2010 = 200 deer per year. The predation rate per deer is 200/200 = 1.0 per year, which is the per capita predation pressure. However, the model should use the functional response: predation = aDW, where a is the attack rate. Here, aW = 1020/200 = 1.0, so a = 0.05. But the logistic growth without predation: rD(1 - D/K) = 0.5200(1 - 200/500) = 100*(0.6) = 60 deer per year. Then subtract predation: 60 - 200 = -140 deer per year. So the deer population is declining at 140 deer per year. But wait, the predation of 200 deer per year is greater than the growth, so the deer would go extinct quickly. However, the wolf population might decline if deer decline. The question asks for the current growth rate, so it's -140 per year. But we must check if the carrying capacity is relevant: the deer are at 200, below K, so food is not limiting; predation is the main factor. So the current growth rate is negative, -140 deer/year. This seems drastic, but it's what the numbers give. Perhaps the predation rate is too high; in reality, wolves might not consume 10 deer per year when deer are scarce. But based on the given, the deer population is declining rapidly. The manager should reconsider the culling.
2. How will the culling of 50 deer per year affect the deer and wolf populations over time? Consider both direct and indirect effects.
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Culling 50 deer per year adds an additional mortality of 50 deer/year. Combined with the current predation of 200 deer/year, total mortality becomes 250 deer/year, while the growth is 60 deer/year, leading to a net decline of 190 deer/year. This would accelerate the deer population decline, potentially causing extinction within a few years. The wolf population, which depends on deer, would then decline due to reduced food availability. The culling might temporarily reduce overgrazing, but it could also destabilize the predator-prey cycle. A better approach might be to reduce the wolf population or to implement a more sustainable culling rate that considers the predator-prey dynamics. For example, culling should be less than the surplus production; currently, the deer population is already declining due to predation, so culling would be counterproductive. The manager should first assess whether the deer population is actually overabundant; at 200 deer, it's below carrying capacity, so overgrazing might not be an issue. The proposed culling seems inappropriate.