For uniform nearest-neighbor bonds of strength $K_0$ on a ring with $N\ge 3$, retain the normalization of the given model:with indices interpreted modulo $N$. On an open chain the interior equations are the same, whileFor nonuniform bonds, replace each $K_0$ by the corresponding $K_{ij}$. A convention without the factor $1/N$ is obtained by rescaling the bond strengths; it is not silently substituted here.

For all-to-all coupling define the Kuramoto order parameterIt is the centroid of unit phase vectors: $r$ measures phase coherence and $\psi$ the mean phase. Multiplying by $e^{-i{\theta }_i}$ and taking imaginary parts givesThus only the two components of the Kuramoto order parameter mediate interaction.

For the symmetric stationary continuum branch, choose the rotating frame with $\psi =0$. Oscillators with $|\omega |\le Kr$ lock at $\theta =\arcsin (\omega /(Kr))$, choosing $\cos \theta >0$ for stability. Those with $|\omega |>Kr$ drift, with stationary phase density proportional to $|\omega -Kr\sin \theta {|}^{-1}$. Their real contribution to the order parameter vanishes, since the integral of $\cos \theta /(\omega -Kr\sin \theta )$ over a period is the change of a periodic logarithm. Their imaginary contributions cancel between $\omega$ and $-\omega$ because $g$ is symmetric.

The locked oscillators therefore give the real self-consistency relationSet $\omega =Kr\sin \theta$ and divide by $r>0$ to obtainThis is the Kuramoto self-consistency equation for partial synchronization. The incoherent state $r=0$ must be considered before this division. If $g$ is continuous at zero with $0<g(0)<\infty$, the branch meets incoherence at $K_c=2/[\pi g(0)]$.

There is always the fixed point $(0,0)$. For $P>0$, the first fixed-point equation gives $e^{-P}=1/r$, and the second gives $P=N(1-1/r)$. Thus for $r>1$ there is exactly one positive fixed point,For $r=1$, every point $(N,0)$, $N\ge 0$, is fixed. There are no other nonnegative fixed points.

At the origin the Jacobian matrix has eigenvalues $r,0$. It is asymptotically stable for $0<r<1$ and a saddle for $r>1$. For $r<1$, the bounds $N_{t+1}\le r{N}_t$ and $P_{t+1}\le N_t$ also prove global extinction in the nonnegative quadrant.

At the positive fixed point,Here $\log r<r-1$ and $\log r>1-1/r$ for $r>1$. The discriminant is negative because the positive trace is less than two and determinant exceeds one. The conjugate multipliers have modulus $\sqrt{N_{\ast }}>1$, so the positive fixed point is always unstable. This is the instability of the Nicholson-Bailey equilibrium.

For completeness, on the $r=1$ fixed line,For $0\le N<1$, linear perturbations are bounded but have a neutral component along the fixed line; there is no asymptotic stability of an individual point. For $N>1$ there is transverse instability. At $N=1$, the repeated unit eigenvalue has a nontrivial Jordan block, so powers have an unbounded off-diagonal entry and the linearized system is unstable. This degenerate case is not covered by a distinct-eigenvalue assumption.

A positive equilibrium must have $v=u=s$. LetIt is the unique positive root of $(1-s)(s+d)=as$. At this point the Jacobian matrix isConsequentlyIt is linearly asymptotically stable when $b>L$, and unstable when $0<b<L$. In either region it is a node or focus according as $(L-b{)}^2-4bsD/(s+d)$ is nonnegative or negative. At $b=L>0$ the linear eigenvalues are purely imaginary. If $L\le 0$, it is stable for every $b>0$.

The requested boundary follows from $1-d-2s=a-D$ and $s/(s+d)=(1-s)/a$:

To prove a limit cycle exists in the unstable case, a compact trapping set must stay away from the singular boundary $u=0$. The following construction supplies it. Choose $M>1$ large enough to contain the positive equilibrium and the orbit being considered. On $u=M$, $\dot{u}<0$, and on $v=M$ with $u\le M$, $\dot{v}\le 0$. Put $P=u{e}^{-v}$. Using $u+d\ge d$ and completing the square in $v$ givesChoose $\epsilon >0$ so small that $\epsilon e^M$ times the square bracket is less than $1/2$, and then choose $0<\eta <\epsilon$. The setis compact in the positive quadrant and forward invariant. On the curved boundary $P=\epsilon$, $u\le \epsilon e^M$ and $\dot{P}>0$. On $v=\eta$, $u\ge \epsilon >\eta$ gives $\dot{v}>0$. Its other boundaries were checked above. The constants can be chosen so the equilibrium is interior.

When $b<L$, the equilibrium is a hyperbolic repeller. Remove a sufficiently small outward-flow neighborhood of it from $\mathcal{K}$. This leaves a compact trapping region containing no equilibria. The Poincaré-Bendixson theorem gives a periodic omega-limit orbit of a nonequilibrium trajectory, hence a limit cycle. The analytic vector field excludes an accumulating family of periodic return-map zeros unless the whole local return map is the identity, which is incompatible with the repelling equilibrium and the nonperiodic approaching trajectory. This is the positive trapping construction for the Leslie-Gower model.

The Poincaré-Bendixson theorem states that the omega-limit set of a forward orbit confined to a compact region of the plane, containing no equilibrium points, is a periodic orbit.

Put $x=R\cos \theta$, $y=R\sin \theta$. Direct calculation givesHere $1/2\le H\le 17/16$ and $|\cos \theta {\sin }^3\theta |\le 3\sqrt{3}/16$. Thus on the inner circle $R=\sqrt{3}/2$ the radial velocity is positive, and on the outer circle $R=3/2$ it is negative. In their closed annulus,so there is no equilibrium. The theorem gives a periodic orbit in this trapping annulus.

It is asymptotically stable: the vector-field divergence isthroughout the annulus. Its nontrivial Floquet multiplier is $\exp {\int }_0^T\mathrm{div}Fdt<1$. This proves an attracting periodic orbit exists. There can be only one there: two such nested orbits would enclose an annular region with zero outward flux, contradicting the strictly negative divergence via the divergence theorem.

Chemotaxis supplies positive feedback: a local increase in bacterial density produces more attractant, whose gradient draws in more bacteria. Ordinary cell diffusion opposes aggregation. Chemical diffusion spreads the signal and removes very sharp gradients, while decay and logistic growth stabilize the uniform kinetics.

The dispersion relation from the preceding matrix makes the competition quantitative: the positive $D_n{D}_a{k}^4$ term stabilizes sufficiently short wavelengths, whereas the negative chemotactic contribution $-\alpha \chi k^2$ can destabilize intermediate wavelengths. Increasing cell diffusivity or chemical decay raises the attraction threshold. A growing allowed spatial mode produces correlated patterns in cell density and chemical concentration; the eigenvector relation $(\lambda +\beta +D_a{k}^2)\delta a=\alpha \delta n$ puts their peaks in phase for the growing real mode.

This predicts initial pattern formation. Nonlinear saturation and stability of the eventual pattern require the nonlinear equations, beyond the linear instability test.

The nonzero homogeneous equilibrium is $\boxed{n_{\ast }=1,a_{\ast }=\alpha /\beta }$. Without transport its Jacobian matrix iswith negative eigenvalues $-1,-\beta$.

Zero chemical flux gives $a_x=0$ at the endpoints; zero bacterial flux then gives $n_x=0$. Linear perturbations therefore have Neumann boundary conditions and modes $\cos (k_{m}x)$, $k_m=m\pi$. For each mode, the amplitude matrix isIts trace is always negative. Its determinant isA mode grows exactly when $\Delta (k_m)<0$. In particular, any nonzero mode is destabilized by sufficiently large $\chi$. For continuously variable $k$, a nonempty unstable band exists precisely whenWriting $C=\alpha \chi -D_a-\beta D_n$, its squared-wavenumber endpoints are $[C\pm \sqrt{C^2-4\beta D_n{D}_a}]/(2D_n{D}_a)$. On the finite interval, at least one allowed $k_m^2$ must lie inside this band. The uniform $m=0$ mode stays stable. This is the chemotactic pattern-instability criterion.

Choose any positive chemical concentration scale $A_0$ and setAfter dropping hats, substitution giveswhereFor explicit dimensional checks, write bacterial-density and chemical-concentration units as $\mathsf{N},\mathsf{A}$. Then $[D_N]=[D_A]=L^2/T$, $[r]=[\mu ]=T^{-1}$, $[\bar{\chi }]=L^2/(T\mathsf{A})$ and $[\sigma ]=\mathsf{A}/(T\mathsf{N})$. Since $[K]=\mathsf{N}$ and $[A_0]=\mathsf{A}$, each displayed ratio has unit one. The new spatial domain is $[0,1]$.

Let $r$ be bacterial intrinsic growth rate, $K$ carrying density, $\sigma$ chemical production per bacterial density per time, and $\mu$ chemical decay rate. The bacterial flux iscombining random diffusion with chemotaxis up the chemical gradient. Chemical flux is $J_A=-D_A{A}_x$. The two local balances give the Keller--Segel model with logistic growth:All transport and rate parameters are positive. The production rate is constant per cell, so total production is proportional to $N$.

The travelling-wave reduction of a reaction-diffusion system of Fisher's equation isMultiplying by $U^{\prime }$ and integrating gives $c{\int }_{\mathbb{R}}(U^{\prime }{)}^{2}dz=1/6$, so the decreasing invasion front has $c>0$. The linearization at $U=0$ has spatial eigenvaluesFor $0<c<2$ they are complex: approaching zero spirals in the $(U,U^{\prime })$ plane and changes the sign of $U$, which is inadmissible for a density. Thus $c\ge 2$ is necessary.

It is also sufficient. Write $V=U^{\prime }$ and choose $\ell =(c-\sqrt{c^2-4})/2>0$. The triangleis forward invariant for $U^{\prime }=V$, $V^{\prime }=-cV-U(1-U)$. On $V=0$, $V^{\prime }\le 0$. On $V=-\ell U$,The boundary $U=1$ points left, and the boundary at $U=0$ consists only of the origin. The unstable branch from the saddle $(1,0)$ enters this triangle, where $U$ decreases and remains nonnegative. Its limiting state must have $V=0$ and $U(1-U)=0$; since it has left $U=1$, it tends to $(0,0)$. Therefore a monotone positive front exists for every $c\ge 2$, including the double-root endpoint:This is the Fisher-KPP minimum wave speed, obtained with an explicit invariant-region existence argument.

Let $J$ be population flux. For any fixed control volume $V\subseteq \Omega$, mass conservation with local production givesAssume local, isotropic random dispersal, no imposed advection, and constant diffusion coefficient $D$. Fick's first law is $J=-D\nabla u$. The divergence theorem and arbitrariness of $V$ yieldThe coefficient has units length squared per time; $f$ has units population density per time. Smoothness sufficient to pass from the integral balance to the local equation is understood. This is a reaction-diffusion equation.