ODE | Phase lines
ODE | Phase lines

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DE2: (CORE) can find equilibria of autonomous differential equations and classify their stability using graphical (phase-line) and algebraic (using the derivative) approaches.
The number of fish N in a fishery grows according to the autonomous differential equation:
dN/dt = 2N, where N is the number of fish and t is in weeks.
1. Find the equilibria of the system (there are 2 equilibrium points).
2. Give an interpretation of each equilibrium point in terms of the fishery system.
3. Classify the stability of each equilibrium using a graphical approach (i.e., phase-line diagram). Your answer should agree with Q4.
4. Classify the stability of each equilibrium using an algebraic approach (i.e., using the derivative). Your answer should agree with Q3.
5. Interpret the stability of each equilibrium point in terms of the fishery system.

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02:37

The population for a fish species in a fish farm obeys differential equationdP dtP(3-P) h, 2+PP(0) = Po,t 2 0,where h is the rate of harvesting, which is constant; and which satisfies 0 < h < 0.5.Find the equilibrium points of the model: (ii) Sketch typical solution curves of P against for different values of P(O), including the equilibrium points. (iii) Draw phase line for the equation. (iv) Determine the nature ofthe equilibrium points (i.e. whether each is stable or unstable): Describe the solution behaviour of P(t) for P(0) > 0. Interpret you result for part by explaining what this means in terms of your model.

01:31

Reaction-diffusion equation: Consider the salmon population in the fishery. We assume that salmon is produced with rate Î» and is removed due to fishing with a rate Î¼. Consider the partial differential equation:

âˆ‚u/âˆ‚t = 2u^2 – 25u + u^2

subject to the boundary conditions:

0 < x < 1, t > 0

where u represents the population level of the salmon. We study this model on 0 < x < 1 with homogeneous Neumann boundary conditions (i.e. âˆ‚u/âˆ‚x(0) = âˆ‚u/âˆ‚x(1) = 0).

Determine the system of two ODEs which describe the steady-state solutions of the partial differential equation.

Find all equilibrium points (steady states) of the system of ODEs and classify each of the equilibrium points into one of six classes (stable node, unstable node, saddle, stable spiral, unstable spiral, center) using linear stability analysis. If the classification of any equilibrium point cannot be determined, please explain why.

03:06

With time \$t\$ in years, the population \$P\$ of fish in a lake (in millions) grows at a continuous rate of \$r \%\$ per year and is subject to continuous harvesting at a constant rate of \$H\$ million fish per year.(a) Write a differential equation satisfied by \$P\$.(b) What is the equilibrium level of \$P\$ if \$r=5\$ and \$H=15 ?\$(c) Without solving the differential equation, explain why the equilibrium you found in part (b) is unstable.(d) Find a formula for the equilibrium for general positive \$r\$ and \$H\$(e) Are there any positive values of \$r\$ and \$H\$ that make the equilibrium stable?

08:14

11.population of fish in a lake satisfies the growth equation: dx/dt = f(x,h) = 0.5x(4 – x) -h, where x(t) is thousands of fish in the lake at time t (in years) and h is thousands of fish harvested per year: If the harvesting term h is zero, how many fish will the lake support (i.e, what is its carrying capacity)? (b) If the harvesting term is constant at h =1, find all equilibrium solutions and draw a phase line. Label each equilibrium as a sink, source, or node. If h = 1 and the initial condition is x(0) = 0.5, what happens to the solution aS t L 0? Explain this in terms that a biologist might use. (d Sketch phase lines for h = 0, 0.5, 1.0, 1.5, 2.0, 2.5 and place them side by side as in Figure 2.23 to form a bifurcation diagram (e) What is the bifurcation point h = h* for this problem?

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