Using differential eq. to model population gowth?

Question

First, we're told to look at this formula for population growth:

N(t) = Ne^(kt)

Where N and k are positive constants. Okay, so that formula works sometimes, but overshoots as we increase year. So here's my problem:

a) A slightly more sophisticated model for population growsh is:

N(t) = Nc /(1 + e^(-bt))

for some positive constants, Nc and b

--Identify the major difference between this and the former model. What is the physical meaning of the constant, Nc?

Thanks all

Answer 1

FIRST MODEL

With the first model, we observe that;

As t→∞ , we see that N(t)→∞

dN/dt= kNe^(kt)

As t→∞ , we see that dN/dt→∞, with the implication that the rate of population growth gets exponentially larger.

The features we observe include;

- Population grows to an infinite size

- Population grows at an increasingly faster rate

The two features above are clearly invalid within an environment; when the population grows to a certain size, the population size should decrease due to constraining factors such as food supply, predation, etc...

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SECOND MODEL

In the second model, we note that;

N(t)

= Nc /(1 + e^(-bt))

As t→∞ , we see that;

e^(-bt)→0

1 + e^(-bt)→1

Therefore N(t)→Nc

dN/dt

= [-b * Nc * e^(-bt)]  /  [1+ e^(-bt)]^2

As per previously, we note that as t→∞;

e^(-bt)→0

1 + e^(-bt)→1

Therefore    dN/dt→0

As t→∞ , we see that dN/dt→0, with the rate of growth getting increasingly slower, until the rate of growth virtually becomes zero, with population remaining constant.

In the second model, we note the following characteristics;

- Population reaches a constant of Nc

- Population growth virtually reaches zero as time approaches infinity

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CONCLUSION

The second population growth model is more realistic, as it takes into account the factors of the environment. The physical meaning of Nc is what we refer to as the 'carrying capacity' of the environment. That is, when the population exceeds the carrying capacity of the environment, it will fall until it reaches Nc, a horizontal asymptote. Likewise, when the population is below the carrying capacity of the environment, it will rise until it reaches Nc.