Structure-Preserving Numerical Schemes for Two-Stage Local and Nonlocal Dispersal Systems

Contributions

• The habitat-conforming dispersal operator has positive off-diagonal entries, preserves constants and discrete mass, is self-adjoint and dissipative, and has only constant vectors in its nullspace.Source: Contributions · p. 1
• For each fixed δ > 0, the habitat-conforming operator is spatially consistent, with an explicit error bound in terms of the approximated function’s modulus of continuity.Source: Contributions · p. 2
• The semi-implicit Euler scheme is uniquely solvable for every Δt > 0 and preserves nonnegative population densities and semidiscrete equilibria.Source: Contributions · p. 2
• For every positive time step, the scheme exactly preserves the sign of the semidiscrete persistence–extinction threshold.Source: Contributions · p. 2
• The paper derives residual-based a posteriori lower and upper bounds for the continuous principal spectrum point, using positive approximate eigenfunctions.Source: Contributions · p. 2
• On rectangular habitats, the reflected discretization gives second-order threshold convergence in h and δ and finite-time error of order Δt + h^2 + δ^2, uniformly in h/δ.The constants are independent of the ratio between mesh size and interaction scale.Source: Contributions · p. 2

2.1 Continuous system, coefficients, and ordering

• The model distinguishes two stages with nonnegative reaction coefficients, strictly positive b and f, nonzero r and s, positive dispersal rates, and nonnegative cross-stage competition.Source: §2.1 · p. 2

2.2 Nonlocal dispersal on the habitat

• For δ > 0, the difference-form nonlocal dispersal operator uses a symmetric strictly positive kernel; the local alternative is the Laplacian with homogeneous Neumann boundary conditions.The general-habitat analysis imposes no local-limit scaling on the nonlocal kernel family.Source: §2.2 · pp. 2–3
• Kernel symmetry gives constant preservation for the continuous dispersal operator and yields the stated dispersal identities.Source: §2.2 · p. 3

2.3 Linearization and persistence threshold

• The principal spectrum point is a spectral supremum because the infinite-dimensional spectral bound need not be attained; the cited continuous-model result associates λδ ≤ 0 with extinction and λδ > 0 with persistence [3, Theorem 1].Source: §2.3 · p. 3

3 Habitat-conforming spatial discretization

• On a conforming cell partition, the discretization uses cell averages and pairwise cell-interaction rates to define the habitat-conforming dispersal operator.Source: §3 · pp. 3–4
• For every δ > 0, the operator has positive off-diagonal entries and annihilates constant vectors; its nullspace consists exactly of constants.Source: §3 · p. 4
• The dispersal equation preserves discrete total mass, a consequence of the operator’s symmetric interaction coefficients.Source: §3 · p. 4
• For fixed δ, spatial consistency is bounded using the modulus of continuity; for Lipschitz functions, the stated bound is O(h).Source: §3 · pp. 4–5

4.1 Nonnegativity, fixed points, and the discrete threshold

• For every Δt > 0, the semi-implicit step has a unique solution and preserves nonnegativity; nonzero data become strictly positive.The discrete maximum principle establishes these properties for the component linear solves, and Theorem 4.3 gives unconditional nonnegativity for the two-stage scheme.Source: §4 · p. 5, §4.1 · p. 6
• Nonnegative fixed points of the scheme are exactly steady states of the semidiscrete system, so the time step does not shift its equilibria.When τ = 0, the semidiscrete system has a unique equilibrium; stability of a hyperbolic, locally asymptotically stable equilibrium carries to the scheme for sufficiently small Δt.Source: §4.1 · pp. 6–7
• For every Δt > 0, ρ(Gδ,h(Δt)) > 1, and hence U = 0 is a linearly unstable fixed point of (25).Source: §4.1 · p. 7

4.2 Computable bounds for the continuous threshold

• A positivity-preserving reconstruction of an approximate eigenfunction gives an a posteriori interval for the continuous principal spectrum point, supporting sign determination.Substituting the reconstructed function and discrete threshold into the stated bounds produces the interval.Source: §4.2 · p. 7

5.1 Reflected nonlocal diffusion

• Coordinatewise even reflection with a compactly supported kernel, positive near the origin, recovers the Neumann Laplacian with an O(δ^2) rate.The reflected continuous operator converges to Δv with homogeneous Neumann boundary condition.Source: §5 · p. 7

5.2 Cartesian discretization

• The reflected Cartesian operator has structural properties including positive coordinate-neighbor interactions and a constant-vector nullspace.Its fixed-δ discretization is consistent with the reflected operator as the mesh is refined.Source: §5.2 · pp. 9–10

5.3 Local-limit consistency

• Under the stated smoothness and even-extension assumptions, discrete local-limit consistency has an error bound independent of h/δ.The proof treats both the local formula and the reflected nonlocal formula; cancellation of the scale factor keeps the bound uniform in their ratio.Source: §5.3 · pp. 10–11

5.4 Box thresholds and asymptotic compatibility

• The reflected Cartesian scheme inherits the nonnegativity, strong-positivity, principal-pair, and threshold arguments for admissible mesh and interaction scales.This transfer uses strictly positive reaction coefficients at cell centers and the coordinatewise maximum-principle argument.Source: §5.4 · pp. 11–12
• The continuous and discrete thresholds converge to their local counterparts, and the threshold approximation error is bounded independently of the relative decay of h and δ.The sign of the discrete threshold agrees with the local threshold for sufficiently small h and δ, independently of their relative rate of decay.Source: §5.4 · pp. 12–13
• For regular local solutions, the nonlinear scheme converges in finite time with a bound whose constant is independent of h/δ.The paper states first-order accuracy in time and second-order accuracy in mesh size and interaction scale.Source: §5.4 · p. 13

6.1 Smooth nonrectangular habitat

• On the smooth elliptical habitat, the discrete principal eigenvalue changes sign as the reproduction coefficient is scaled, without rectangular geometry or even reflection.Table 1 presents the discrete threshold for that habitat.Source: §6.1 · p. 13, Table 1 · p. 14

6.2 Rectangular habitat and local-limit convergence

• The reflected nonlocal threshold approaches the local discrete threshold with an observed log–log slope of approximately 2.08 on a fixed fine grid.The reported error measures convergence of λ□δ,h on that grid.Source: §6.2 · p. 14, Table 2 · p. 14

6.3 Rectangular threshold dynamics and nonnegativity

• The negative-threshold run has λ□δ,h = −0.1361 and mean total density 2.57 × 10−3 at t = 30; the positive-threshold run has λ□δ,h = 0.2697 and approaches a positive fixed point.All computed components remain nonnegative in both runs.Source: §6.3 · p. 14, Figure 2 · p. 15, Figure 3 · p. 16

7.1 Habitat-conforming scheme

• For every Δt>0, the semi-implicit update has a unique solution and preserves nonnegativity.The proof uses the blockwise maximum-principle structure of the linear system.Source: §7.1 · pp. 16–17
• A positive threshold gives spectral radius greater than one, a negative threshold gives spectral radius less than one, and a zero threshold gives spectral radius exactly one.The positive and negative cases are bounded respectively above one and below one; in the zero case, one is an eigenvalue and the spectral radius equals one.Source: §7.1 · p. 18
• For a positive threshold, the semiflow has a unique equilibrium; a hyperbolic, locally asymptotically stable semidiscrete equilibrium remains locally asymptotically stable for the numerical scheme when Δt is sufficiently small.The equilibrium conclusion follows from strong monotonicity, boundedness, and strong sublinearity; the stability statement applies under the stated hyperbolicity and stability assumption.Source: §7.1 · p. 19

7.2 Reflected Cartesian local-limit results

• Under Assumption 5.6, the reflected-scheme consistency residual is bounded through times T with a constant independent of h/δ.The local consistency argument combines dispersal, time-difference, and interstage-term estimates.Source: §7.2 · p. 21
• The reflected-scheme finite-time error estimate uses ε = Δt + h^2 + δ^2 and holds uniformly over the time steps up to T.The proof subtracts the consistency identity from the numerical step and bounds the resulting error equation.Source: §7.2 · p. 22

• On general smooth habitats, the habitat-conforming discretization preserves discrete mass, dissipation, nonnegativity, semidiscrete equilibria, and the persistence–extinction threshold for each fixed interaction scale.Source: §8 · p. 22
• On rectangular habitats, the reflected discretization gives second-order threshold convergence in h and δ and finite-time solution error of order Δt + h^2 + δ^2, with constants independent of h/δ.Thus these local-limit estimates require no prescribed coupling between mesh size and interaction scale.Source: §8 · p. 22
• The reflection condition (43) makes explicit the fourth-order boundary compatibility used in the local-limit estimates.Source: §8 · p. 22
• Numerical experiments confirm the analyzed conservation, threshold, nonnegativity, and local-limit behavior.Source: §8 · p. 22