Question

a) You have an exponentially-growing population of rodents, with an annual λ of 1.35.

Calculate the number of individuals added to the population in a given month when there are 250, 500, 1000, 5000, or 10,000 individuals to start with. (2pts)

b) Now impose density dependence on the population with a logistic model in which K = 6500 individuals. Again, calculate the number of individuals added to the population in a given month when there are 250, 500, 1000, 5000, and 10,000 individuals to start with. (2pts)

c) How does population growth compare between the two models (exponential vs. logistic) at the low and at the high starting population numbers? (2pts)

Present the numbers (rounded to two decimal places) for a), and b) in a table in the space below, along with the answer for c).

Answer 1

From relationship for exponential growth of population N_t = N₀*e^(lambda)t......Eq 1.

where t= 1 month = 1/12 y = 0.083 y, lambda = 1.36, and N_t and N₀ are population at the end of 1 month and initial population respectively.

a) Putting the value of initial population N₀ as 250, 500, 1000, 5000, and 10, 10000, we get. N_t as 279, 560, 1119, 5595, and 11190. Therefore, increase in respective populations or no. of individuals added to the population in a given month are 29, 60, 119, 595 and 1190 respectively.

b) On imposing density dependence with a logistic on the population in which K (carrying capacity) = 6500 individuals. We replace lambda by (K - N₀)/K in Eq 1. Our new equation will be

N_t = N₀*e^((K - N₀)/K)t......Eq 2. Again putting N₀ as 250, 500, 1000, 5000, 10000 we get N_t as 271, 540, 1072, 5101, 5827. Therefore,s no. of individuals added to the population in a given month would be 21, 40, 72, 101, and -4173 (No. of individuals in this population will decrease).

Please put the nos. in the Table as directed in the question.

c) As you see by imposing density dependence on the model and taking carrying capacity as 6500 individuals. The no. of individuals added at low starting population almost remain the same in both exponential growth and logistic growth (29 Vs 21) whereas no. of individuals at high starting population actually went down from their original number in logistic growth model (logically they must remain below carrying capacity, K all the time).

Thanks!