You are trying to predict the future population size five years from now of a population of 500 bison that has an intrinsic growth rate of 0.10 per year, but you are not certain of the carrying capacity of the ecosystem

General
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you are trying to predict the future population size five years from now of a population of 500 bison that has an intrinsic growth rate of 0.10 per year, but you are not certain of the carrying capacity of the ecosystem. what is the difference in your predictions if the carrying capacity of the ecosystem is 1,500 bison compared to if the carrying capacity is 750 bison?

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Answer:

To predict the future population size of bison, we need to use the exponential growth model. The formula for exponential growth is:

P(t) = P(0) * e^(r * t)

Where:
P(t) = population size at time t
P(0) = initial population size
e = Euler’s number (approximately 2.71828)
r = intrinsic growth rate
t = time in years

In this case, we have an initial population size of 500 bison and an intrinsic growth rate of 0.10 per year. We want to predict the population size five years from now.

If the carrying capacity of the ecosystem is 1,500 bison, we assume that the population will eventually stabilize at this level. The carrying capacity represents the maximum number of individuals that the ecosystem can support sustainably. To incorporate this into our prediction, we can modify the exponential growth model to include a limiting factor:

P(t) = K / (1 + (K - P(0)) / P(0) * e^(-r * t))

Where:
K = carrying capacity

Using this modified formula, we can calculate the predicted population size five years from now with a carrying capacity of 1,500 bison:

P(t) = 1500 / (1 + (1500 - 500) / 500 * e^(-0.10 * 5))
= 1500 / (1 + 2 * e^(-0.50))
≈ 1036.27

Therefore, if the carrying capacity of the ecosystem is 1,500 bison, the predicted population size five years from now would be approximately 1036.

Now, let’s consider the case where the carrying capacity is 750 bison. Using the same modified formula, we can calculate the predicted population size:

P(t) = 750 / (1 + (750 - 500) / 500 * e^(-0.10 * 5))
= 750 / (1 + 1 * e^(-0.50))
≈ 585.68

So, if the carrying capacity of the ecosystem is 750 bison, the predicted population size five years from now would be approximately 586.

In summary, the difference in predictions between a carrying capacity of 1,500 bison and 750 bison is that the population size five years from now would be higher (around 1036) with a higher carrying capacity and lower (around 586) with a lower carrying capacity. The carrying capacity directly affects the population’s ability to grow and reach its maximum sustainable size.