Question

I need help with these please. The equations my professor has given are: The equations my professor has given are:

d’=d+cN

dN/dt=rN(1-(N/K)) —logistic growth equation:

#2)

equation:

Nt=K/{1+[(K-N0)/N0]e^-rt}

r=0.21

K=2400

N0=420

1. For low density sea otter populations in the North Pacific Ocean, the overall annual per capita birth rate (b) is about 0.6 and the death rate is about 0.47 . Birth rate is not density-dependent but death rate is density dependent, such that d
′
=d+cN (where N= number of otters per km of coastline). If c=0.0133, what is the carrying capacity sea otter density in number of animals per km of coastline? [5pts]

2. The intrinsic rate of increase ( r ) for northern elephant seals about 0.21 (per annum). We will assume that population regulation is linearly density dependent. [8 pts]

a. If the carrying capacity of elephant seals at a recently recolonized island is 2400 adult females and the initial population size is 420, how many more years (approximately) will it take for the population to grow to K ? "hint: you will need to project population growth several years into the future
b. If the island population overshot carrying capacity to 3300 adult females, how many more years (approximately) will it take for the population to decline to K ? *hint: you will need to project population growth several years into the future
Expert Answer

Answer 1

The carrying capacity for sea otters is determined by equating the per capita birth rate to the density-dependent death rate, while several years of population growth for elephant seals are projected using the logistic growth equation to estimate when the population will reach its carrying capacity.

Step-by-step explanation:

Estimating Population Carrying Capacity

For sea otters, the carrying capacity can be determined using the provided death rate equation d' = d + cN. Since the birth rate (b) is not density-dependent, we can equate birth and death rates to find the carrying capacity when the population growth rate is zero: b = d + cN. Solving for N when b=0.6 and d=0.47 gives us the carrying capacity for sea otters per km of coastline.

Logistic Growth in Elephant Seals

Using the logistic growth equation, Nt = K / {1 + [(K - N0) / N0] * e^-rt}, where Nt is the population size at time t, K is the carrying capacity, N0 is the initial population size, r is the intrinsic rate of increase, and e is the base of the natural logarithm, we can project the growth over several years to estimate when the population will reach K. For the initial population of 420 and K=2400 with r=0.21, we can calculate the number of years required for the population to grow to its carrying capacity. Similarly, for population decline from an overshoot, we use the same logistic growth model but start with N0=3300.