Consider the differential equation . for x20, which models a population of fish with harvesting that...
Consider the differential equation for , which models a population of fish with harvesting that is governed by the logistic equation and a number of fish H caught and removed every unit of time (harvesting). Here the parameters r, K, and H are all positive. a) Assume that . Draw the phase plane. b) Assume that . What happens to the population of fish as t increases? Can I have a step-by-step walkthrough on how to solve these two. I...
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,...
A population of fish is modeled by the logistic equation modified to account for harvesting ??/?? = .25? (1 − (?/4) ) - a or equivalently ??/?? = − (1/16) ?^2 + (1/4) ? - a 1.) What will happen to the size of the population in the long run? If your answer depends on the initial population size (hint – it should!), be sure to include all possible scenarios. 2.) Use MATLAB to solve this differential equation. (Yes, it...
A population of fish is modeled by the logistic equation modified to account for harvesting: de = 25p (1-) -a or equivalently top2 + 3p-a 1. What are the equilibrium solutions? Note: your answer(s) should be expressed in terms of a 2. Determine the range of values of a for which this model has 2 different equilibrium values. When does it have 1 equilibrium? O equilibria? Carefully interpret each of these scenarios in the context of this model. 3. Suppose...
What are the solutions to the following diff eq’s? A fish hatchery employed a mathematician to design a model to predict the population size of fish that the hatchery can expect to find in their pond at any given time. The mathematical model that the mathematician created is: dP dt 25 (a) Draw a one dimensional phase portrait of the autonomous differential equa- tion. What does this differential equation predict for future fish populations for various initial conditions? Describe 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...
#2 Consider the following model for the dynamics of a population of size N (measured as number of individuals x 10) over time (in months) that is subject to harvesting: The population grows according to a logistic equation in the absence of harvesting and h is a constant per a) Find all equilibria and determine the values of h for which each is stable or unstable. 4aestng andcnstant capita harvest rate. b) Construct the bifurcation plot: plot the equilibria from...
Exercise 4: Assume that a population is governed by a logistic equation with carrying capacity K intrinsic growth rate r, and initial population size K is subjected to constant effort harvesting: (a) Determine the population size, N(t) (b) Verify that if E< r, the population size will approach the positive steady state, Ni, the carrying capacity K if Erand if E>r, the population will approach the zero steady state, No, astoo. (c) Find the maximum sustainable yield of the population.
I need help with question #3 When there is no fishing, the growth of a population of clown fish is governed by the following differential equation: dy dt 200 where y is the number of fish at time t in years. 1. Solve for the equilibrium value(s) and determine their stability. Create a slope field for this differential equation. Use the slope field to sketch solutions for various initial values. 2. 3. Summarize the behavior of the solutions and how...
Exercises 1. Verify equation (3) 2. Use the techniques of Section 13.7 and the fact that P(0) = 10 to solve equation (5). 3. The carrying capacity of Atlantic harp seals has been estimated to be C = 10 million seals. Let 1 = 0 correspond to the year 1980 when this seal population was estimated to be about 2 mil- lion. (Data from: Fisheries and Oceans Canada.) (a) Use a logistic growth model = kP(C - P) with k...