Initially the populations fit the logistic equation but over time, the populatio
ID: 102503 • Letter: I
Question
Initially the populations fit the logistic equation but over time, the populations seem to drift away from K or the asymptote. For larger animals, the results were even less supportive. Can you speculate why this may have been so? Thus, it is very hard to find examples that fit the logistic equation but it is a good null model to go by. We will see that time lags are probably one of the elements to predicting or modeling population growth. Robert May understood the limitations of the logistic model and showed how variation in simple deterministic population growth alter the behavior of population growth. He looked at time lags. If there is a time lag of length tau between a change in the size of a population and its effect on the population growth rate, then the population growth timer t is controlled by its size at some time in the past, f - tau. If we incorporate the time lag into the alternative form of logistic growth equation (on left), then the time lag affects the population growth. To see this, first consider an unlagged population, for which the alternative form of the logistic equation is dN/dt = rN(1 - N_1/K). Then if r =1.1, K = 1, 000, and N = 900, what do we have? So what is the new population size? If we incorporate a time lag such that at the time the population is 900 the effects of crowding are being felt a though the population were only 800, then what would the new population size be then? So what is the effect of time lag on the growth compared to no time lag?Explanation / Answer
Answer:
i) For unlagged population, the logistic equation for logistic growth in continous time is: dN/dt = rN(1 - Nt/K)
Where,
For given r = 1.1, K = 1000, N = 900
dN/dt = 1.1*900*(1-900/1000) = 99
ii) With incorporation of time lag:
N(t -T) = population size at time t-T in the past = 800
dN/dt = rN(1 - Nt-T/K)
dN/dt = 1.1*900*(1-800/1000) = 198
iii) Effect on time lag on the growth compared to no time lag:
After the introduction of time lag, the change in population size increases.