2. A two-sex seahorse population growth model A process of modeling is to critically reexamine any model you have come up with, discover its deciencies, and continually improve it. It is obvious that the common Malthusian model on population growth is decient: dN = BN DN, dt where N (t) is the population density, B is the per capita birthrate, and D is the per capita deathrate. It is decient because it seems to imply that any member of the population can bear ospring! Obviously, this is not true most species we're familiar with have two biological sexes, only one of which gives birth. Typically, the female sex bears ospring. A notable exception is the seahorse: It is actually the male seahorse that bears the ospring and gives birth! Just for fun, let's suppose we're modeling the population dynamics of seahorses. A simple modication to our population model to take account of the fact that only male seahorses give birth, might be dN = By DN, (3) dt where y(t) is the population density of the male seahorses and N (t) = x(t) + y(t) is the total seahorse population, with x(t) the population density of female seahorses. However, this model also has a deciency: it predicts seahorse births even if there are no females in the population! This is appropriate only if the males are the limiting species (i.e., y x). If the females are the limiting species (x y), the births should be proportional to x: dN = Bx DN. (4) dt We would like to have a model that works for any x and y densities and reduces to Eq. (3) if y x and Eq. (4) if x y. One such model is dN xy =B DN. dt x+y This model is symmetric with respect to the two sexes. Since there are two unknowns, however, we need to split this equation up into two equations, one for each of x and y: dx xy = Bx dt x+y dy xy = By dt x+y Dx x, (5) Dy y, (6) where Dy is the male per capita death rate and Dx is the female per capita death rate of the seahorses; By is proportional to the male birthrate and Bx is proportional to the female birthrate. Use Eqs. (6) and (5) to answer the following questions. a. (5 points) Find the nontrivial equilibrium male and female seahorse populations (x , y ). It turns out that in general there is no nontrivial equilibrium unless certain conditions involving the parameters are met. Find that condition. b. (5 points) Even if x and y cannot reach equilibrium individually (if the condition in a. is not satised), there is a well-dened equilibrium for the sex-ratio. Calculate the equilibrium sex-ratio (x/y) . Assume that the death rates, Dx and Dy , are the same. i.e., Dx = Dy = D. c. (10 points)For Dx = Dy = D, show that 1 1 1 y(t) x(t) = y(t0 ) By Bx By 1 x(t0 ) e Bx D(t t0 ) , for initial conditions y(t0 ) and x(t0 ). d. (10 points) From c it is seen that after a few generations (with a generation dened by D(t have Bx x(t) y(t). By Substitute this into Eq. (6) and solve for y(t). 2 t0 ) 1), we practically