Final answer:
Equilibrium in a population subject to overcompensation occurs when the growth rate is zero, resulting in no change in population size year over year. The equilibria for the given model are found by solving the equation xt = 5xt/(1+xt²), yielding two solutions: xt = 2 and xt = 0. However, only xt = 2 represents a realistic population size (200 individuals), as population cannot be negative.
Step-by-step explanation:
To determine the equilibria for a population described by the equation x₁ = 5xt/(1+xt²) where xt represents the population number in year t (measured in hundreds of individuals), we need to set the population number in the next year equal to the population number in the current year. This implies setting x₁ equal to xt, yielding the equation xt = 5xt/(1+xt²). At equilibrium, the population does not change from one year to the next, which is a condition satisfied when the growth rate equals zero.
The solution to xt = 5xt/(1+xt²) is found by multiplying both sides of the equation by (1+xt²) to get xt + xt³ = 5xt. We can subtract xt from both sides to get xt³ = 4xt. Now, we can divide both sides by xt, assuming xt is not zero, to obtain xt² = 4, which means xt = 2 or xt = -2. However, since xt represents a population size and cannot be negative, the only physically meaningful solution for the equilibrium is xt = 2 (which corresponds to 200 individuals, as the measurement unit is hundreds of individuals). There is also an equilibrium at xt = 0, which represents the extinction or absence of the population.
Given that real-world populations cannot sustain exponential growth indefinitely due to limited resources and other density-dependent factors, the logistic model incorporating carrying capacity provides a more realistic description of population dynamics. Overcompensation in population dynamics represents scenarios where high population levels can lead to a decrease in the population size due to factors such as resource depletion or increased predation.