52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.


where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.


d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).

Verified step by step guidance

1. Identify the differential equation model given for the population \(P(t)\), which is typically the logistic growth model: \[\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right),\] where \(r > 0\) is the growth rate and \(K > 0\) is the carrying capacity.
2. Understand that the equilibrium solutions occur when the growth rate is zero, i.e., when \[\frac{dP}{dt} = 0.\] This happens if either \(P = 0\) or \(P = K\).
3. Analyze the stability of these equilibrium points: since \(r > 0\), \(P = 0\) is an unstable equilibrium and \(P = K\) is a stable equilibrium, meaning the population tends to \(K\) as \(t \to \infty\).
4. Therefore, to find \[\lim_{t \to \infty} P(t),\] recognize that the solution \(P(t)\) approaches the stable equilibrium \(K\) over time.
5. Finally, verify that this limit is consistent with the graph in part (c), which should show the population leveling off at the carrying capacity \(K\) as time increases.

Key Concepts

Logistic Differential Equation

The logistic differential equation models population growth with a carrying capacity, expressed as dP/dt = rP(1 - P/K). Here, P(t) is the population at time t, r is the growth rate, and K is the maximum sustainable population. Understanding this equation helps analyze how populations grow and stabilize over time.

Limit of a Function as t Approaches Infinity

The limit lim(t→∞) P(t) describes the long-term behavior of the population. Evaluating this limit reveals the steady-state or equilibrium population size. In logistic growth, this limit typically equals the carrying capacity K, indicating population stabilization.

Equilibrium Solutions and Stability

Equilibrium solutions occur when the population does not change over time (dP/dt = 0). For the logistic model, P = 0 and P = K are equilibria. Stability analysis shows that P = K is stable, meaning the population tends to this value as t increases, consistent with the graph's behavior.