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A Let P(t) denote the population size at time t. We are given that the growth rate of the population is proportional to population size with proportionality constant k. Therefore, the growth rate at time t is kP(t).
The death rate is constant and equals a organisms per unit time.
The rate of change of the population size is given by the derivative . We also know that the change in population size comes from births and deaths. So the rate of change of population size is the birth rate - death rate. Writing this in mathematical notation gives us the required differential equation:
. The given initial population size is the initial condition for this equation.
Combining the two, we find the model for our population as
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B Now let us try to solve the given differential equation .
If we write this equation in standard form, we find
The method to solve this type of first order equations is to multiply by an integrating factor and then integrating.
The integrating factor can be found as . Multiply the equation throughout by this integrating factor.
We can rewrite the equation again as
Now we can integrate both sides with respect to t to find
Multiply throughout by to find the expression for P(t).
Next we apply the given initial condition P(0) = Po to find the constant C.
Substituting for C gives us the final solution
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C In this part, we look at the asymptotic behavior of the solution.
We are given or .
With this in mind, let us look at the solution.
The first term in the solution is a constant.
The second term has an exponential term. Since it is the proportionality constant for growth rate, k is positive. So the term as . This exponential term is also multiplied by a constant . We were given that this constant is also positive, so the entire term goes to infinity as t does.
We found the behavior of each term in the solution as . Combining what we found
We have found that
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Do the question completely. Especially part C thanks 4. A population P of organisms dies at a constant rate of a or...
please complete the whole question 4. A population P of organisms dies at a constant rate of a organisms per unit time, and the growth rate is proportional to the population size with the proportionality constant k A. Assume the initial population P(0) = Po. Write a differential equation that models the size of the population P(t) at ay time t. B. Write the equation from part A in standard form, and solve. (The answer must contain the terms Po,...
&7 4. A population P grows at a constant rate of a organisms per unit time, and the death rate is proportional to the population size with the proportionality constant k. A. Assume the initial population P(0) Po. Write a differential equation that models the size of the population P(t) at ay time t. B. Write the equation from part A in standard form, and solve. (The answe terms Po, a, k and a constant C.) wer must contain the...
Part B Please!! Scenario The population of fish in a fishery has a growth rate that is proportional to its size when the population is small. However, the fishery has a fixed capacity and the growth rate will be negative if the population exceeds that capacity. A. Formulate a differential equation for the population of fish described in the scenario, defining all parameters and variables. 1. Explain why the differential equation models both condition in the scenario. t time a...
Population Growth: Let P(t) be the number of rabbits in the rabbit population. In the simplest case we can assume the number of rabbits born at any moment of time is proportional to the number of rabbits at this moment of time. Mathematically we can write this as a differential equation: Here b is the birth rate, i.e. births per time unit per rabbit. In the model above we ignore deaths and assume resources are unlimited. A. Solve the equation...
step by step please 4. Suppose that the logistic equation dt Pla -bP) models a population of fish in a lake after t months during which no fishing occurs. What is the limiting population for this fish population? suppose that, because of fishing, fish are removed from the lake at a rate proportional to the existing fish population. i. Write a differential equation that describes this situation. ii. Show that if the constant of proportionality for the harvest of fish,...
2. The growth rate of a population of bacteria is directly proportional to the population p() (measured in millions) at time t (measured in hours). (a) Model this situation using a differential equation. (b) Find the general solution to the differential equation (c) If the number of bacteria in the culture grew from p(0) = 200 to p(24) = 800 in 24 hours, what was the population after the first 12 hours?
Q2- Fish Population In this question, we will use differential equations to study the fish population in a certain lake. An acceptable model for fish population change should take into account the birth rate, death rate, as well as harvesting rate Let P(t) denote the living fish population (measured in tonnes) at time t (measured in year) Then the net rate of change of the fish population in tonnes of fish per year is P'(t): P'(t) birth rate - death...
1a) If the rate of change, W, is inversely proportional to W(x) where W is always greater than 0. If W(1)=25 and W(4)=17, then find W(0) 1b)assume an object's weight, W, is proportional to its height, H. Does it make sense to use differential equations to model the Weight of the object? If so, write the differential equation. If not, explain why 1c)Assume the population, P, of cats in a region is proportional to the area, A, of the region....
One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. (a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.) dy (b) Solve the differential equation. Assume y(o) (c) A small town has 2100 inhabitants. At 8 AM, 100 people have heard a...
Urgently need the answers. Please give right answers. Q2 Fish Population In this question, we will use differential equations to study the fish population in a certain lake. An acceptable model for fish population change should take into account the birth rate, death rate, as well as harvesting rate. Let P(t) denote the living fish population (measured in tonnes) at time t (measured in year) Then the net rate of change of the fish population in tonnes of fish per...