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Phys. 7224 Practice assignment
Problem 1
Spruce budworm outbreak. The dynamics of spruce budwork population is often modeled by the following PDE
∂tu = ru
u k
u^2 1 + u^2
where u describes the number of budworms per unit area of the forest. The first term on the right-hand-side describes the dynamics of the local density of budworms without predators, with r being the birth rate and k the “capacity” or largest equilibrium density supported by the foliage in the forest, reflecting competition for the food resource. The second term describes the effect of predation (mostly by birds), which saturates at large numbers of budworms (birds can only eat that much). Finally, the last term is dispersal of budworms (in the butterfly stage – caterpillars cannot move from tree to tree).
(a) To begin with, let us switch of diffusion (dispersal), in other words, set D = 0. Show that there is a choice of parameters r and k which leads to four equilibrium solutions. Hint: you might want to plot the first and second terms separately and look for intersection of the two curves. What is the stability of these four equilibria? The stable equilibrium with the smallest u is called the “refuge” regime, u = ur, while the largest to the “outbreak” regime u = uo.
(b) Now let’s turn dispersal on. Does the linear stability of any of the equilibrium states change (assuming a laterally infinite forest)?
(c) Next suppose that the system is laterally finite and for simplicity assume that the forest is one-dimensional, that is the spruce trees grow along the line x ∈ (0, L). The appropriate boundary conditions are u(0) = u(L) = 0 as the budworms need something to eat in order to survive. This boundary condition is only compatible with the uniform state ue = 0, which corresponds to total extinction of the budworm population. Of course, if that state is unstable, u can be nonzero inside the domain. Determine the minimum size L of the system that can support a nonzero population.
(d) Let us now assume that the u(x) ∝ sin(πx/L). The saturated steady state solutions of the evolution equation can then be found approximately by replacing ∂^2 xu → −(π/L)^2 u. Choose some values for r and k which correspond to four equilibria (four zeros of f (u)) and by plotting f (u) and D∂ x^2 u determine graphically the minimal size of the system necessary to support an outbreak level of budworms.
Lotka-Volterra models. Dynamics of interacting populations is often described by simple models such as Lotka- Volterra. For instance, the model of two populations looks like this:
∂tu = u(k + au + bv) + Du∇^2 u, ∂tv = v(n + cu + dv) + Dv ∇^2 v.
(a) Suppose that there is only one species, e.g., v = 0. Interpret the meaning of the constants k and a. By analogy, the constants n and d have the same meaning for the other species. Now suppose both species are present. What is the meaning of constants b and c?
(b) Suppose there is no dispersal of species, Du = Dv = 0. What kinds of equilibrium states are possible for different choices of parameters? In particular, what kinds of steady states exist in the predator-prey system (a, b, d < 0; k, c > 0; n = 0)? How about two competing species (a, b, c, d < 0; k, n > 0)? Is it possible for predators and prey to coexist in a stable state? How about competitors?
(c) Now turn on dispersal. Is this system capable of producing a stationary instability? What are the conditions under which the instability is type-Is? What is the critical wavelength and characteristic time scale? Is type-IIIs instability possible and if yes when? Is type-IIs instability possible? Interpret your conditions in terms of the species interaction (e.g., predator-prey and so on). Is it possible for predators and prey to coexist in a stable state? How about competitors?
(d) Is this system capable of producing an oscilatory instability? What are the conditions under which the instability is type-Io? What is the critical wavelength and characteristic time scale? What is the critical frequency? Is type-IIIo instability possible and if yes when? Is type-IIo instability possible? Interpret your conditions in terms of the species interaction (e.g., predator-prey and so on).
Nonlocal coupling coupled neuronal networks. In the examples we have considered so far the evolution has always been described by partial differential equations. Moreover, the dissipation almost always resulted from diffusive- like terms such as ∇^2 u in the evolution equations. In other words, the evolution of the system at some spatial location only depended on the value of the field and its derivetives at that same location. In other words, the dynamics was defined locally. This situation, in fact, does not exhaust all the possibilities. Often the coupling (diffusion, dispersal, mutual influence, etc.) in the system is nonlocal, that is it extends over finite distances. Sometimes the coupling is global, i.e., extends over the whole domain (e.g., pressure affects the dynamics of the fluid globally). In either of these cases the coupling has to be described using integral, rather than derivative, terms, so that one ends up with integral or integro-differential evolution equations. For instance, coupled neurons are often described by models (in one dimension) of the following type:
∂tu(x, t) = f (u) +
w(x, x′)(u(x′, t) − ub)dx′, (1)
where f (u) is a nonlinear function of the firing rate u with one or two stable equilibria describing the dynamics of isolated neurons, w(x, x′) represents the coupling strength, and ub is one of the stable equilibria. In order for the model to be invariant with respect to translations and reflections, one should choose w(x, x′) = w(|x − x′|). Generalization to higher dimensions is straightforward.
(a) There is a close relation between this integral equation and standard partial differential equation in the case of a strongly localized integral kernel w(|x − x′|). By Taylor expanding u(x′, t) in small x′^ − x show that the above integral equation can be rewritten as
∂tu(x, t) = f¯ (u) + D 2 ∂^2 xu + D 4 ∂^4 xu + D 6 ∂ x^6 u + · · · ,
where Dn are some constants. The higher-order diffusion coefficients Dn quickly approach zero as n → ∞. Show that Dn− 1 ∝ σn/n! for large n, where σ is the width of w(x).
(b) The kernel is often chosen in the form of a finite sum of Gaussians
w(x, x′) =
∑^ N
n=
an exp
(x − x′)^2 2 σ^2 n
Do the linear stability of the integral equation (1) for the case N = 1. (Hint: the integral operator commutes with the operator of spatial translations, so the two operators have a common set if eigenfunctions – Fourier modes.) What is the instability type for a 1 > 0? For a 1 < 0?
(c) Redo the linear stability analysis for N = 2, a 1 > 0 and a 2 < 0. Compute the critical wave number qc as a function of an and σn. What is the condition on σ 1 and σ 2 for the instability to be of type-I? Type-III? Is there something in common with the Turing instability? Neuronal networks are often modeled by choosing N = 2, a 1 > 0, a 2 < 0 and σ 1 < σ 2. This type of coupling corresponds to local activation with long range inhibition.
(d) Visual hallucination patterns can be modeled by the following system describing the dynamics of two interacting networks of cells
∂tu= −u + fu(Wuu[u] − Wuv [v]) ∂t v= −v + fv (Wvu[u] − Wvv [v]),
where fu and fv are some “threshold” or sigmoidal functions (something like (1 + tanh x)/2) and
Wkl[y] ≡
wkl(x − x′)y(x′)dx′, wkl(x) = αkl exp
x^2 2 σ^2 kl
Perform the linear stability analysis and compare the results with those for Turing instability.
Complex Ginzburg-Landau equation. Consider the following equation
τ 0 ∂tA = è(1 + ic 0 )A + (1 + ic 1 )ξ 02 ∇^2 A − (1 − ic 3 )g 0 |A|^2 A,
where τ 0 , ξ 0 , g 0 ,^ c 0 , c 1 and c 3 are real constants (parameters) and A(x, t) is a complex field.
(a) Show that the parameter c 0 can be eliminated by redefining the field A to obtain the equation
τ 0 ∂t A˜ = è A˜ + (1 + ic 1 )ξ^20 ∇^2 A˜ − (1 − ic 3 )g 0 | A˜|^2 A.˜
(b) Next show that the equation can be nondimensionalized to obtain
∂T A¯ = A¯ + (1 + ic 1 )∇^2 X A¯ − (1 − ic 3 )| A¯|^2 A.¯
This last equation is known as the complex Ginzburg-Landau equation (CGLE). When c 1 → ∞ and c 3 → ∞ the CGLE reduces to the nonlinear Scr¨odinger equation which describes, for instance, propagation of light in optical fibers. On the other hand, when c 1 → 0 and c 3 → 0 the CGLE reduces to the Ginzburg-Landau equation which describes, for instance, the magnetization inside magnetic domains in ferromagnetics.
(c) Conduct the linear stability analysis of the CGLE and determine the instability type. (Be careful, the field A¯ is complex!)
(d) Find the nonlinear saturated solutions consistent with the type of instability.
(e) If you feel inspiration, try to conduct the linear stability analysis of the saturated solutions and show that a secondary instability (called a Benjamin-Feir instability) occurs for c 1 c 3 = 1.
