Question

¹ Let's suppose that the bird population in a certain continent has a growth rate of 2%. (This means the population grows proportional to the total population and the constant of proportionality is 0.02 .) Suppose the population is 4 million at this point in time, which we'll call t=0. Due to the environmental situation, there is a change in the bird population in the region. New birds are entering the country at a rate of 100,000 per year but also there is a population decline at a rate of 200,000 per year. Let P=P(t) be the population in millions at time t, measured in years.

(a) Write a differential equation reflecting the situation. Keep in mind that P is in millions.
(b) Do a qualitative analysis of this differential equation.
(c) Solve this differential equation to find an expression for P(t).
(d) If this situation goes on indefinitely, what will happen to the country's bird population? What does the differential equation tell you about how the population is changing at time t=0? What will happen as time goes on, meaning what is limₜ→[infinity]​
P(t)? Answer this in two ways.
i. using part (b);
ii. using part (c).
(e) What is the bird population after 2 years, according to the solution to the model?
(f) Will the bird population ever reach 2 million? If so, when? If not, why not?

Answer 1

To solve the bird population dynamics, we create a linear differential equation, perform qualitative analysis to understand the long-term behavior, and solve the equation to find P(t). The analysis predicts the population will stabilize, and we can calculate the population at any given time and determine if it will reach certain thresholds.

Step-by-step explanation:

Population Dynamics and Differential Equations

For the bird population problem described, the differential equation incorporating the growth rate and the influx and decline of the bird population can be expressed as: dP/dt = rP + I - O, where r is the per capita growth rate (0.02), I is the immigration rate (0.1 million birds per year), and O is the outflow, including deaths (0.2 million birds per year). The initial population (P0) is 4 million birds.

(a) The differential equation reflecting the bird population dynamics is therefore: dP/dt = 0.02P + 0.1 - 0.2. Converting the bird numbers to millions for uniformity.

(b)The qualitative analysis indicates that the population will eventually reach a steady state where the growth rate and the net migration rate (immigration minus emigration) balance each other out.

(c)To solve the differential equation, integrate to get: P(t) = Ce^(0.02t) + (0.1-0.2)/0.02, where C is the constant of integration that can be determined using the initial value P(0) = 4. After simplifying the equation, we find the expression for P(t).

(d)Using the qualitative analysis (part b) and the solution to the differential equation (part c), we can infer that the bird population will asymptotically approach a population size where the net migration rate is zero. This is because the exponential growth will balance out with the net migration, leading to a stable population size.

(e) To find the bird population after two years, substitute t=2 into the derived expression for P(t) from part c and calculate the population size.

(f) To determine whether the population will ever reach 2 million, we need to examine the limits of the solution for P(t). If the limit implies the population size decreases indefinitely, then a population size of 2 million will not be reached. Otherwise, setting P(t) to 2 and solving for t will give the timeframe when the population reaches 2 million, if it is mathematically possible.