Competition-Diffusion Lotka–Volterra Systems

• Competition-diffusion Lotka–Volterra systems are reaction-diffusion models that couple local competitive interactions with spatial dispersal to analyze coexistence, exclusion, and bistability.
• The models employ spectral thresholds and traveling-wave analysis to determine conditions under which species can persist, invade, or be excluded.
• Extensions incorporating delays, memory, nonlinear diffusion, and stochastic forcing enrich the analysis by capturing realistic ecological and spatial complexities.

Searching arXiv for recent and foundational papers on competition-diffusion Lotka–Volterra systems. A competition-diffusion Lotka–Volterra system is a spatially extended form of competitive population dynamics in which the densities of competing species evolve under local Lotka–Volterra interactions and spatial dispersal. In its classical PDE form, it couples diffusion with intra- and interspecific competition, often in a bounded habitat with homogeneous Dirichlet or Neumann boundary conditions, and its central questions concern persistence, exclusion, coexistence, stability of steady states, and invasion through traveling fronts. In spatially heterogeneous environments, dispersal rates and resource profiles become part of the dynamical threshold structure, and global asymptotic stability of semitrivial and coexistence steady states becomes a principal object of analysis (Wang, 2021).

1. Mathematical formulation and model classes

The standard heterogeneous two-species competition-diffusion Lotka–Volterra system has the form

$\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - a_{11}(x)u - a_{12}(x)v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - a_{21}(x)u - a_{22}(x)v\big), \end{cases}$

with nonnegative initial data and either homogeneous Dirichlet or homogeneous Neumann boundary conditions. Here $d_1,d_2>0$ are diffusion coefficients, $m_i(x)$ are intrinsic growth-rate functions, $a_{ii}(x)$ are intraspecific competition rates, and $a_{ij}(x)$ for $i\neq j$ are interspecific competition rates. A common simpler form assumes constant competition coefficients,

$\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$

In this framework, spatial heterogeneity means that the local environment varies with position $x$, most importantly through $m_1(x)$ and $m_2(x)$, and sometimes also through the competition coefficients. Larger $d_1,d_2>0$0 means species $d_1,d_2>0$1 disperses more rapidly across space.

Several structurally important variants appear in the literature. One special triangular system has

$d_1,d_2>0$2

so that the evolution of $d_1,d_2>0$3 is independent of $d_1,d_2>0$4; this makes the full dynamics reducible to scalar logistic theory plus principal spectral analysis (Chen et al., 2020). Another strand uses finite-dimensional patch systems rather than a Laplacian. In the two-patch almost periodic model,

$d_1,d_2>0$5

“diffusion” means conservative inter-patch dispersal between two discrete habitats, not continuous-space intra-patch diffusion (Muhammadhaji et al., 2013). This suggests that the expression “competition-diffusion” is used both narrowly for reaction-diffusion PDEs and more broadly for competitive dispersal systems.

2. Steady states, spectral thresholds, and global dynamics

The standard steady states are the trivial state $d_1,d_2>0$6, semitrivial states of the form $d_1,d_2>0$7 and $d_1,d_2>0$8, and coexistence states $d_1,d_2>0$9 with both components positive. The semitrivial state $m_i(x)$0 is determined by the scalar logistic elliptic problem

$m_i(x)$1

and similarly for $m_i(x)$2. Under a standard sign convention, a positive scalar steady state exists if and only if the principal eigenvalue $m_i(x)$3 of

$m_i(x)$4

satisfies $m_i(x)$5. Different papers define the principal eigenvalue with opposite sign, so one must check conventions carefully.

The global-dynamics classification typically replaces the algebraic inequalities of the ODE Lotka–Volterra competition model by spectral invasibility criteria. If neither species can persist alone, one expects extinction of both species. If $m_i(x)$6 exists and species 2 cannot invade it, namely

$m_i(x)$7

then competitive exclusion by species 1 occurs under suitable hypotheses. If each species can invade when rare,

$m_i(x)$8

then coexistence is typically possible, and under appropriate uniqueness assumptions the positive coexistence steady state is globally asymptotically stable. If both semitrivial states are locally stable, bistability may occur, and the winner depends on initial conditions. The standard picture is therefore extinction, exclusion, coexistence, or bistability, with principal eigenvalues governing the thresholds (Wang, 2021).

In the triangular heterogeneous system

$m_i(x)$9

the scalar logistic equation for $a_{ii}(x)$0 is autonomous, and the full dynamics are determined by the sign of

$a_{ii}(x)$1

If the semitrivial steady state $a_{ii}(x)$2 is not linearly unstable, then it is globally asymptotically stable and there is no positive coexistence steady state; if it is unstable, then the system has a unique positive coexistence steady state, and that coexistence state is globally asymptotically stable (Chen et al., 2020).

3. Traveling waves, spreading speeds, and invasion structure

On the whole line, the one-dimensional Cauchy problem and the traveling-wave ansatz

$a_{ii}(x)$3

or $a_{ii}(x)$4, depending on convention, lead to a second major branch of the theory: invasion fronts, spreading speeds, and large-time profile selection. In delayed $a_{ii}(x)$5-species competition-diffusion systems with distributed delays,

$a_{ii}(x)$6

positive invasion waves from $a_{ii}(x)$7 to the coexistence equilibrium exist for every

$a_{ii}(x)$8

provided there is an instantaneous self-limitation component $a_{ii}(x)$9, and under

$a_{ij}(x)$0

the waves converge to the positive equilibrium as $a_{ij}(x)$1; the same framework also yields nonexistence below the threshold and allows nonmonotone waves (Lin et al., 2013).

For the classical two-species strong-competition system on $a_{ij}(x)$2,

$a_{ij}(x)$3

the two pure states $a_{ij}(x)$4 and $a_{ij}(x)$5 are stable, and the dynamics are organized by a unique bistable front with speed $a_{ij}(x)$6. In the scenario where $a_{ij}(x)$7 is native and $a_{ij}(x)$8 invades successfully, the solution converges to a translated bistable wave with exact asymptotic speed $a_{ij}(x)$9. If both species are invasive, two qualitatively different regimes occur. When the scalar KPP speed of $i\neq j$0,

$i\neq j$1

is larger than the scalar KPP speed of $i\neq j$2,

$i\neq j$3

the rightward front is asymptotically the scalar Fisher–KPP front of $i\neq j$4, with Bramson correction $i\neq j$5, and $i\neq j$6 uniformly on the right half-line. When $i\neq j$7, a propagating terrace forms: a leading KPP front of $i\neq j$8 moving at speed $i\neq j$9 with correction $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$0, and behind it a bistable replacement front of speed $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$1 in which $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$2 displaces $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$3 (Peng et al., 2019).

In the strong-weak regime

$\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$4

the minimal speed $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$5 of the wave connecting $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$6 to $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$7 can be nonlinearly determined. In the pushed case

$\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$8

sharp Cauchy asymptotics are still available. If $\begin{cases} u_t = d_1 \Delta u + u\big(m_1(x) - b_1 u - c_1 v\big),\[2mm] v_t = d_2 \Delta v + v\big(m_2(x) - c_2 u - b_2 v\big). \end{cases}$9 invades a habitat already occupied by $x$0, the solution converges to the minimal pushed traveling wave with speed $x$1. If both species start from compact support and $x$2, then under the condition

$x$3

the asymptotic state splits into two fronts: a leading scalar KPP front of $x$4 and a trailing pushed competition front moving at speed $x$5 (Wu et al., 2022).

The critical competition case

$x$6

is qualitatively different because the associated ODE has the whole segment $x$7 of equilibria. In the PDE

$x$8

there is no ultimately monotone traveling wave connecting two distinct equilibria on that line. Yet the Cauchy problem with compactly supported data still exhibits sharp selection: if $x$9, then $m_1(x)$0 excludes $m_1(x)$1 and converges to its scalar KPP front with logarithmic shift $m_1(x)$2, while if $m_1(x)$3, the symmetric statement holds for $m_1(x)$4. The refined interior profile reveals a new bump phenomenon in which the loser and the defect $m_1(x)$5winner decay with heat-kernel order in sublinear regions (Alfaro et al., 2021).

Weak-competition traveling waves also support genuinely nonmonotone structures. In the strict weak competition regime

$m_1(x)$6

traveling fronts from $m_1(x)$7 to

$m_1(x)$8

exist for all

$m_1(x)$9

and explicit sufficient conditions are given for nonmonotone waves in either component. In the critical weak competition case

$m_2(x)$0

the paper proves existence of front-pulse traveling waves (Chen et al., 6 Oct 2025). Entire solutions, defined for all $m_2(x)$1, add a global-in-time layer to this theory: new entire solutions have been constructed that satisfy

$m_2(x)$2

and then evolve into regime-dependent layered patterns such as

$m_2(x)$3

in weak competition and

$m_2(x)$4

in strong competition (Lam et al., 2020).

4. Mobility, advection, and the sign of propagation

A recurrent question in competition-diffusion theory is whether higher mobility is advantageous. The answer is not uniform across models. In a symmetric two-species reaction-diffusion framework with a standing front, a small diffusion asymmetry changes the front speed by

$m_2(x)$5

where $m_2(x)$6 is the adjoint translational mode. For a toy model, the sign of $m_2(x)$7 can change with a parameter, so more mobility may be either advantageous or disadvantageous. For the symmetric Lotka–Volterra competition-diffusion model

$m_2(x)$8

the near-onset result is different: when $m_2(x)$9, increasing the mobility of one species gives that species an invasion advantage at first order (Risler, 2017).

In the strong-competition traveling-wave problem, the sign of the wave speed itself is the ecological indicator of which species prevails. Recent work focuses on the symmetric regime in which the species differ only through diffusion. In the normalized wave system

$d_1,d_2>0$00

with $d_1,d_2>0$01, $d_1,d_2>0$02, $d_1,d_2>0$03, the sign convention is that $d_1,d_2>0$04 implies species $d_1,d_2>0$05 displaces $d_1,d_2>0$06. Near the strong-competition borderline $d_1,d_2>0$07, the zero-speed profile satisfies

$d_1,d_2>0$08

and there exists $d_1,d_2>0$09 such that

$d_1,d_2>0$10

This verifies the “Unity is not strength” conjecture in that parameter regime (Chen et al., 3 Jul 2026).

A complementary study gives explicit sufficient conditions for the sign of the unique bistable wave speed in the classical strong-competition system

$d_1,d_2>0$11

In the symmetric case $d_1,d_2>0$12, $d_1,d_2>0$13, the paper proves that the wave speed is negative if either

$d_1,d_2>0$14

or

$d_1,d_2>0$15

where

$d_1,d_2>0$16

Here the sign convention is reversed from the previous paper: $d_1,d_2>0$17 corresponds to successful invasion by $d_1,d_2>0$18. This difference of convention is itself standard in traveling-wave literature and must be checked case by case (Nakamura et al., 16 Oct 2025).

Directed dispersal enriches the picture further. In the advection model

$d_1,d_2>0$19

species $d_1,d_2>0$20 moves away from regions of high competitor density $d_1,d_2>0$21. Large enough $d_1,d_2>0$22 can destabilize the homogeneous coexistence equilibrium, generate nonconstant positive steady states by bifurcation, and in a large-advection regime produce a shadow system and transition-layer solutions modeling species segregation (Gai et al., 2014).

5. Delay, memory, nonlinear diffusion, and stochastic forcing

Several extensions replace instantaneous, linear dispersal by memory, delay, nonlinear transport, or stochastic perturbation. In the memory-based diffusion competition model

$d_1,d_2>0$23

with Dirichlet boundary conditions, there exists a branch of spatially nonhomogeneous positive steady states. The paper establishes stability of that branch under weak memory or suitably opposing memory effects and proves Hopf bifurcation as memory delay increases. Strong memory may destabilize the coexistence state and lead to periodic oscillations, while spatial heterogeneity enriches the dynamical behaviors (Li et al., 20 Apr 2025).

The delayed reaction-diffusion framework with distributed delays reaches a different conclusion. If each self-interaction kernel has an instantaneous atom $d_1,d_2>0$24, then large distributed delays in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling waves. More precisely, positive invasion waves exist for every

$d_1,d_2>0$25

and the asymptotic convergence to the coexistence equilibrium is governed by

$d_1,d_2>0$26

not by smallness of the delay tail itself (Lin et al., 2013).

Nonlinear diffusion and cross-advection also alter the well-posedness theory. In the advective Lotka–Volterra model

$d_1,d_2>0$27

global bounded classical solutions exist under explicit growth conditions relating the nonlinear diffusion exponent $d_1,d_2>0$28, the sensitivity growth exponent $d_1,d_2>0$29, and the damping exponent $d_1,d_2>0$30. For the fully parabolic system one sufficient condition is

$d_1,d_2>0$31

while in a parabolic-elliptic attractive case the advection term becomes coercive and attraction may inhibit blow-up (Wang et al., 2016).

Stochastic competition-diffusion systems introduce multiplicative environmental randomness. On $d_1,d_2>0$32 with Neumann boundary conditions, the SPDE model

$d_1,d_2>0$33

admits a unique global mild solution, preserves nonnegativity and positivity, has spatial-temporal Hölder regularity, admits one-point densities under Malliavin calculus assumptions, and possesses an invariant measure under suitable nondegeneracy hypotheses. The paper also gives an extinction criterion: if

$d_1,d_2>0$34

then

$d_1,d_2>0$35

and similarly for $d_1,d_2>0$36 (Nguyen et al., 2020).

6. Discrete patches, multispecies extensions, and related systems

The two-patch almost periodic competition model shows that the competition-diffusion theme extends beyond PDEs. Under the assumptions that the coefficients are bounded, continuous, nonnegative, and almost periodic, and that diffusion is not too strong relative to intrinsic growth,

$d_1,d_2>0$37

together with inequalities such as

$d_1,d_2>0$38

the system has a unique almost periodic positive solution, and every positive solution converges to it. This is a discrete dispersal analogue of global attractivity in continuous-space competition-diffusion models (Muhammadhaji et al., 2013).

For $d_1,d_2>0$39-species systems,

$d_1,d_2>0$40

the theory becomes substantially harder when $d_1,d_2>0$41. The literature explicitly notes that coexistence of all species or of subsets of species, as well as complex spatio-temporal patterns and oscillations, may occur, while no known tools provide a complete analytical study in general. Instead, rigorous exclusion criteria are available in extreme invasion regimes. If the exotic species $d_1,d_2>0$42 has sufficiently large $d_1,d_2>0$43 and directly harms each native species, then the invasion is always successful and the native species are driven to extinction; if $d_1,d_2>0$44 is sufficiently small, then the invasion always fails, and under additional equal-diffusivity assumptions the invader converges uniformly to zero (Contento et al., 2018).

Related predator-prey systems can sometimes be analyzed through competition-diffusion reductions. In the modified Leslie–Gower model with diffusion in heterogeneous environments,

$d_1,d_2>0$45

comparison with triangular competition-diffusion systems yields global stability of semitrivial states in some parameter ranges and uniform persistence with at most one positive coexistence steady state in others (Chen et al., 2020).

Taken together, these results present the competition-diffusion Lotka–Volterra system not as a single model but as a family of dispersal-competition systems whose common mathematical core is the interaction between diffusion or dispersal, local competitive kinetics, and the geometry of steady states and invasion fronts. The principal analytic themes are spectral thresholds, comparison structures, traveling-wave selection, and the way spatial heterogeneity, mobility asymmetry, delay, memory, and noise reorganize the classical alternatives of exclusion, coexistence, and bistability.