### Lotka-Volterra Predator-Prey Model**Problem Statement:**A predator-prey interaction is described by the Lotka-Volterra model:\[ x' = 0.3x + 0.04xy \]\[ y' = 0.4y - 0.04xy \]**Tasks:**(a) **Find the critical point in the first quadrant.**\[ (x, y) = \left( \_\_\_ , \_\_\_ \right) \]**Numerical Solution and Population Cycles:**Use a numerical solver to sketch some population cycles. (Choose \( x(0) = 9 \) and \( y(0) = 6 \)).**Graph Descriptions:**- **Top Left Graph:** - `x(t)` in blue exhibits oscillatory behavior peaking around 12 and dropping to 6. - `y(t)` in red oscillates with peaks around 8 and troughs near 4.- **Top Right Graph:** - `x(t)` in blue oscillates with peaks around 10 and troughs near 8. - `y(t)` in red exhibits smoother oscillations peaking near 6 and dipping to 4.- **Bottom Left Graph:** - `x(t)` in blue shows regular oscillations with peaks around 12 and troughs near 6. - `y(t)` in red exhibits a more prominent oscillation peaking around 10 and troughs near 4.- **Bottom Right Graph:** - `x(t)` in blue exhibits irregular oscillatory behavior, dropping significantly at certain points. - `y(t)` in red also shows irregular oscillations dropping to lower values.(b) **Estimate the period of the periodic solutions that are close to the critical point in part (a).**(Choose \( x(0) = 9 \) and \( y(0) = 6 \). Use a CAS to estimate the period and round your answer to two decimal places.)\[ \text{Estimated Period:} \_\_\_\_ \]

Answered: Image /qna-images/answer/09c638fd-ab80-48bb-b235-fc960fc00eda.jpg

Answered: A predator-prey interaction is…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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1) For the DDS given by Yn = 1/5 ( Yn-1)^2 + 6/5. Find the equilibrium values. Write your answers as a list, separated by commas. 1a) Ye = ?
K Solve the following initial value problem. y (3) - 2y + y'=6+7xe*; y(0) = y'(0) = 0, y''(0) = 1 The solution is y(x) =
(M1.1 A) Consider the Berries on a Farm model, where the bushels of berries increase by 10% every month and 600 bushels are harvested every month. For this work we will use At =1. Here's the model: 1 B' = -В - 600 10 B(0) = Bo. If B_0=3000, what happens to this model as the months go by? If B_0=9000, what happens to the madel as the months go by? What is the best initial value, B_0 for this model, so that the farm stays stable.
Consider a population P(t) satisfying the extinction explosion equation = ap - bP. where B= ap is the time rate at which births occur and D = bP is the rate at which deaths occur. If the initial population is P(0) = P, and B births per month and D,P. D, deaths per month are occurring at time t=0, show that the threshold population is M= B. dP == ap - bP can be rewritten in the form = kP(M - P) where k= -a and M=- Since, D= bP and B= aP, M can be rewritten in terms of B. D, and P. M-0
(ii) Eliminate the arbitrary constant in the equation y = x² + ₁²x + c₂e³x Answer must be in the form d²y + a₁(x). dxn a₁(x). dn-ly dxn-1 dy + ... an-1(x). + an(x)y = R(x) dx
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
The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem dN = N(1 dt 0.0004N), N(0) = 1. - (a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the new procedure over a long period of time. supermarkets By hand, sketch a solution curve of the given initial-value problem. N 2500 2500 1200 1000 2000 2000 800 1500 1500 600 1000 1000 400 500 500 200 10 15 20 10 15 20 10 15 20 N 1200 1000 800 600 400 200 t 20 10 15 (b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a). N(t) = How many supermarkets are expected to adopt the new technology when t = 10? (Round your answer to the nearest integer.) supermarkets
3.d only chain rule

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