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Assignment 1

The document outlines an assignment involving ecological modeling of snowshoe hare and lynx populations using Lotka-Volterra equations, requiring analysis near equilibrium populations through Taylor expansion. It also includes a section on a one-dimensional potential function, asking for equilibrium points, classification of stability, and computation of angular frequencies. Lastly, it examines the periodic motion of a particle, asking for the time period and the time average of the squared distance from the origin.

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jitanianushka84
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20 views1 page

Assignment 1

The document outlines an assignment involving ecological modeling of snowshoe hare and lynx populations using Lotka-Volterra equations, requiring analysis near equilibrium populations through Taylor expansion. It also includes a section on a one-dimensional potential function, asking for equilibrium points, classification of stability, and computation of angular frequencies. Lastly, it examines the periodic motion of a particle, asking for the time period and the time average of the squared distance from the origin.

Uploaded by

jitanianushka84
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Assignment 1

1. Ecologists have observed cyclical fluctuations in the populations of snowshoe hares (prey) and lynxes (predators)
in the Canadian wilderness. The populations can be modeled by the Lotka-Volterra equations:
dH dL
= 0.1H − 0.01HL, = −0.2L + 0.005HL
dt dt
where: H(t) = Hare population at time t (years), L(t) = Lynx population at time t (years). You will analyze this
system near its equilibrium (constant) populations using Taylor expansion (to linearize the equations):
(i). Find the constant (equilibrium) populations H0 (hares) and L0 (lynxes) where both dH dL
dt = 0 and dt = 0.
(ii). Define small deviations from equilibrium:

h(t) = H(t) − H0 , l(t) = L(t) − L0 .

Use Taylor expansion (ignoring nonlinear terms like h · l) to derive linear differential equations for h(t) and l(t).
(iii). Show that h(t) and l(t) both satisfy the equation:

d2 x
+ ω 2 x = 0,
dt2
and find ω (angular frequency).
(iv). Write the general solutions for h(t) and l(t) in terms of sin and cos. Given initial deviations h(0) = 10 and
l(0) = 0, find explicit expressions for h(t) and l(t).

2. Consider the one-dimensional potential


1 4 a 2
V (x) = x − x ,
4 2
where a is a real parameter.
(a) Compute V ′ (x) and find all equilibrium points as a function of a. Compute V ′′ (x) and classify the equilibria
(stable/unstable) in the two regimes: a < 0 and a > 0. Identify the critical value of a where the qualitative nature of
equilibria changes.
(b) For each stable equilibrium found in part (a), compute the angular frequency ω of small oscillations (in terms of
a).
(c) Sketch the “bifurcation diagram”: equilibrium positions x vs parameter a, showing which branches are stable and
which are unstable.
(d) What happens to the small-oscillation period near a = 0 and why (hint: how does ω behave as a → 0+ ?).

√
3. Motion of a particle is described by the equation (x, y) = (cos ωt, cos( 2ωt)).
(i) Is the motion periodic, and if so what is the time period?
(ii) Define r2 (t) = x2 (t) + y 2 (t). Compute the time average of r2 (t) over an infinitely long time interval:
Z T
1
⟨r2 ⟩ = lim r2 (t)dt.
T →∞ T 0

Can you interpret its value?

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