Consider the model given bydPp² +P- 213Р— 22= -dtwhere P denotes the population size after time t.(i) Determine the equilibrium point (s).(ii) Does the population grow according to Malthusian or logistic model?(iii) Sketch of the phase line.(iv) Use the phase line in to sketch the solution curve of the model.(v) Determine the bound of the population of the given model if the size of the population is null initially.

Answered: Image /qna-images/answer/43361cdc-9aea-4543-b210-bee7123fc4dc.jpg

Answered: Consider the model given by dP 1 3 p2 +…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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2. The DTDS given by P+1 = 3(P; + 18) describes a population in which 18 individuals are added to the population at the beginning of each generation and of that population survives. (a) Starting from a population of Po = 8 individuals, calculate the next three terms, that is, find P1, P2, and P3. (b) Calculate the equilibrium point of the model above. Show necessary algebra!
b) A metal component is subjected to heat-treatment called normalisation. The component is heated to a temperature T, (which is above the critical phase transformation temperatures) and cooled in air. The elementary change dT(t) of temperature T(t) of a heated component is determined from the heath energy balance equation cmdT(t) =-hA(T(t)– Tuir)dt where cmdT(t) is the energy which is lost due to cooling; hA(T(t)– Tuir )dt is the total heat flux from the component to the air during an infinitesimally small time interval dt. The constant c [J/(kg K)] is the specific heath capacity, m is the mass of the component in kg, h [W/(m? K)] is the heat transfer coefficient and A is the surface area in m?. Derive an equation which gives the temperature T(t) as a function of the cooling time t.
For the equation dy/dt=y^4+4y^3-5y^2 (a) Find all the equilibrium solutions and determine their stability type (b) sketch the phase line and draw several solutions curves in the ty plane (c) determine all the inflection levels, if any at all, for the solutions curves
1) Plot the phase line of the following equation and discuss its qualitative implication dy/dt = y2 - y- 6
5
Use a sketch of the phase line to argue that any solution to the logistic model below, where a, b, and po are positive constants, approaches the equilibrium solution p(t) = as t approaches + ∞. 응 dt (a - bp)p; p (to) = Po
3. Solve the initial value problem (explicit solution) t- dt , x(1) = 0.
11. Consider the population model dy dt = 3y - y² - 2. This is a harvesting equation with a harvesting rate of 2. Find the steady states and their stability, and interpret your answer in terms of the population model. Is this overhar- vesting, underharvesting, or critical harvesting?
State the order of the given ordinary differential equation. By(6) 5y" + 5y = 0 Determine whether the equation is linear or nonlinear by matching it with (6) in Section 1.1. ghty + + dx7 dx-1 22 (2) +ao(x)y= g(x) (6) linear O nonlinear Submit Answer
Consider the logistic equation dP dt = -KP(1-B) where k> 0 and B> 0 are constants. (a) Find and classify equilibrium solutions and draw a phase line. (b) Use your work from part (a) to sketch the family of solutions corresponding to this differential equation. (c) If P(t) represents a population that is governed by the reverse logistic equation, use your work in part (b) to interpret the behavior of P(t) and explain the possible fates of such a population. (d) Use your results from part (c) to determine the arbitrary constant if P(0) = Po. Is it possible to evaluate the lim P(t)? If so, evaluate the limit, if not, explain why. t-∞0
What is the phase-line diagram of the function dP/dt = P3- 3P2 - 28P

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