Assuming logistic growth, if I told you there are currently 100 quail on campus, with r=0.1, but that they have greatly exceeded their carrying capacity = K = 50. How many quail will be in the population next generation?
So r is rate of growth, and it determines the increase based on the total 100, but how does carrying capacity factor in?
In the case of logistic growth, the carrying capacity (K) is a limit to the population size that can be sustained by the available resources in the environment. When the population exceeds the carrying capacity, the growth rate (r) slows down and eventually reaches a stable equilibrium.
To calculate the population size in the next generation, we need to use the logistic growth equation:
N(t+1) = N(t) + r * N(t) * (1 - N(t)/K)
where:
N(t+1) is the population size in the next generation
N(t) is the current population size
r is the rate of growth
K is the carrying capacity
In this case, the current population size is 100 (N(t) = 100), the rate of growth is 0.1 (r = 0.1), and the carrying capacity is 50 (K = 50).
Plugging these values into the logistic growth equation:
N(t+1) = 100 + 0.1 * 100 * (1 - 100/50)
N(t+1) = 100 + 0.1 * 100 * (1 - 2)
N(t+1) = 100 + 0.1 * 100 * (-1)
N(t+1) = 100 + (-0.1) * 100
N(t+1) = 100 - 10
N(t+1) = 90
Therefore, the population size in the next generation would be 90 quail.
In logistic growth, the carrying capacity (K) represents the maximum population size that an environment can sustainably support. As the population approaches or exceeds the carrying capacity, the growth rate slows down and eventually reaches equilibrium.
To calculate the population size in the next generation in this scenario, we can use the logistic growth formula:
Nt+1 = Nt + (r * Nt * (1 - (Nt / K)))
Where:
Nt+1 = population size in the next generation
Nt = current population size (100 quail in this case)
r = rate of growth (0.1 in this case)
K = carrying capacity (50 in this case)
Since the current population size has greatly exceeded the carrying capacity, the term (1 - (Nt / K)) will be negative. However, negative growth is not meaningful in this context, so we assume it is zero. This means that the population will not decrease when it exceeds the carrying capacity.
Applying the formula, we get:
Nt+1 = 100 + (0.1 * 100 * (1 - (100 / 50)))
= 100 + (0.1 * 100 * (1 - 2))
= 100 + (0.1 * 100 * -1)
= 100 + (-10)
= 90
Therefore, in the next generation, the population size is estimated to be 90 quail.