Appendix A — Toolkit

This appendix collects the mathematical methods used repeatedly in the module, in the form in which you need them in the exam.

A.1 Solving first-order ODEs

Exponential growth. \(\dot N=rN\Rightarrow N=N_0e^{rt}\).

Linear with constant inhomogeneity. \(\dot N=rN+a\). Shift by the steady state \(N^*=-a/r\): \(M=N-N^*\) satisfies \(\dot M=rM\), so \(N=N^*+(N_0-N^*)e^{rt}\).

Integrating factor. \(\dot M+\mu M=g(t)\). Multiply by \(e^{\mu t}\): \(\frac{d}{dt}(e^{\mu t}M)=e^{\mu t}g(t)\), then integrate.

Separation of variables. \(\dot N=f(N)\): \(\int\frac{dN}{f(N)}=\int dt\). Use partial fractions, e.g. \(\frac{K}{N(K-N)}=\frac1N+\frac1{K-N}\).

Trajectories of planar systems. Along a solution of \(\dot x=f\), \(\dot y=g\): \(\frac{dy}{dx}=\frac{g}{f}\). This is often separable (Lotka–Volterra, SIR).

A.2 Steady states and stability

model steady state stable if
\(\dot N=f(N)\) \(f(N^*)=0\) \(f'(N^*)<0\)
\(N_{t+1}=f(N_t)\) \(f(N^*)=N^*\) \(\lvert f'(N^*)\rvert<1\)
\(\dot{\mathbf x}=\mathbf F(\mathbf x)\) in 2D \(\mathbf F(\mathbf x^*)=0\) \(\operatorname{tr}A<0\) and \(\det A>0\)

Here \(A\) is the Jacobian \(\begin{pmatrix}f_x&f_y\\g_x&g_y\end{pmatrix}\) at the steady state. Simplify its entries using the steady-state equations before computing the trace and determinant.

Classification in 2D. The eigenvalues solve \(\lambda^2-\operatorname{tr}(A)\lambda+\det A=0\).

  • \(\det A<0\): saddle.
  • \(\det A>0\), \((\operatorname{tr}A)^2\ge4\det A\): node (stable if \(\operatorname{tr}A<0\)).
  • \(\det A>0\), \((\operatorname{tr}A)^2<4\det A\): spiral (stable if \(\operatorname{tr}A<0\)).
  • \(\det A>0\), \(\operatorname{tr}A=0\): centre of the linearisation.

For a triangular matrix the eigenvalues are the diagonal entries.

Nullclines. \(\dot x=0\) curves (flow vertical) and \(\dot y=0\) curves (flow horizontal). Steady states are their intersections. Determine the sign of \(\dot x\) and \(\dot y\) in each region.

A.3 Bifurcations

where condition types
1D ODE \(f'(N^*)=0\) saddle-node, transcritical, pitchfork
1D map \(f'(N^*)=1\) saddle-node, transcritical, pitchfork
1D map \(f'(N^*)=-1\) period doubling
2D ODE \(\det A=0\) saddle-node, transcritical, pitchfork
2D ODE \(\operatorname{tr}A=0\), \(\det A>0\) Hopf (limit cycle)

A.4 Non-dimensionalisation

  1. Write each variable as a scale times a dimensionless variable, e.g. \(N=Ku\), \(t=\tau/r\).
  2. Substitute and divide by a common factor.
  3. Choose the scales so that as many coefficients as possible become \(1\).
  4. The remaining dimensionless parameter combinations are the ones that matter.

A.5 Matrices and difference equations

Linear recurrence. \(\mathbf N_{t+1}=L\mathbf N_t\). Try \(\mathbf N_t=\lambda^t\mathbf v\), so \(L\mathbf v=\lambda\mathbf v\). The general solution is \(\sum_ic_i\lambda_i^t\mathbf v_i\), and the dominant eigenvalue determines the long-term behaviour.

Geometric series. \(\sum_{n=0}^{t-1}x^n=\frac{1-x^t}{1-x}\) and \(\sum_{n=0}^\infty x^n=\frac1{1-x}\) for \(|x|<1\).

Leslie matrix. First row \(F_a=b_{a+1}S_a\), subdiagonal \(S_a\). The Euler–Lotka equation is \(\sum_ab_al_a\lambda^{-a}=1\) with \(l_a=S_0\cdots S_{a-1}\).

A.6 Linear PDEs

Separation of variables. Try \(n=f(t)r(a)\) or \(u=X(x)T(t)\), divide by the product, and set both sides equal to a separation constant.

Boundary conditions for \(X''=-\rho^2X\) with \(X=A\cos\rho x+B\sin\rho x\):

at \(x=0\) at \(x=L\) modes
\(X'(0)=0\) \(X(L)=0\) \(\cos(\rho_nx)\), \(\rho_n=\frac{(2n+1)\pi}{2L}\)
\(X(0)=0\) \(X'(L)=0\) \(\sin(\rho_nx)\), \(\rho_n=\frac{(2n+1)\pi}{2L}\)
\(X(0)=0\) \(X(L)=0\) \(\sin(\rho_nx)\), \(\rho_n=\frac{n\pi}{L}\), \(n\ge1\)
\(X'(0)=0\) \(X'(L)=0\) \(\cos(\rho_nx)\), \(\rho_n=\frac{n\pi}{L}\), \(n\ge0\)

Fourier modes for linear stability. \(\xi=Be^{\sigma t}\cos(kx)\) turns \(\partial_x^2\) into \(-k^2\).

Travelling waves. With \(u(x,t)=U(z)\), \(z=x-ct\), we have \(\partial_t=-c\frac{d}{dz}\) and \(\partial_x=\frac{d}{dz}\). At the leading edge, linearise and try \(e^{-\lambda z}\). Real \(\lambda\) gives the minimum speed.

A.7 Taylor expansion

\(f(x^*+h)=f(x^*)+f'(x^*)h+O(h^2)\), and in two variables \(f(x^*+h,y^*+k)=f+f_xh+f_yk+O(2)\). Also useful: \(e^x\approx1+x\) and \(\ln(1+x)\approx x\) for small \(x\).