Three-Species Dynamical Model

• Three-Species Dynamical Model defines a system where three entities interact on a simplex via nonlinear, cubic update rules.
• It employs discrete-time dynamics with parameters (a, b, c) that determine cyclic dominance, extinction, or robust coexistence.
• The model bridges to continuous-time formulation using a variable speed function, offering insights into ecological stability and mathematical behavior.

A three-species dynamical model describes the evolution of systems where three distinct types of entities interact according to prescribed nonlinear rules, often capturing essential features of ecological, chemical, or social systems. The mathematical formulation of such models encapsulates the possible persistence, extinction, or coexistence of the three types, as well as complex phenomena such as cyclic dominance, convergence to fixed points, and the (non-)ergodicity of the resulting system. Discrete-time variants, as considered in (Jamilov et al., 2019), often serve as both qualitative models exhibiting genuinely unique dynamics and as approximations to continuous-time models when appropriately parameterized.

1. Model Definition and State Space

The three-species discrete-time model considered in (Jamilov et al., 2019) operates on the simplex

$$S^2 = \left\{ x = (x_1, x_2, x_3) \in \mathbb{R}^3 : x_i \geq 0, \ x_1 + x_2 + x_3 = 1 \right\}$$

where each coordinate $x_i$ corresponds to the normalized relative abundance of species $i$. This normalization ensures that the total population is constant, and the system evolves inside the triangle defined by the simplex.

Dynamics are defined by an evolution operator $W_{f,a,b,c}$, parameterized by a continuous speed function $f : S^2 \to (0,1]$ and interaction constants $a, b, c$ (chosen in the open interval $(−1,1) \setminus \{0\}$), which modulate the strengths and signs of the nonlinear interactions among the species.

2. Mathematical Formulation

The update rules for the discrete-time map are specified as: $$\begin{aligned} x_1' &= x_1 [1 + (a x_1 x_2 - b x_3) f(x)] \ x_2' &= x_2 [1 + (c x_2 x_3 - a x_2) f(x)] \ x_3' &= x_3 [1 + (b x_3 x_1 - c x_2) f(x)] \ \end{aligned}$$ or more compactly,

$x_i' = x_i [1 + \Phi_i(x) f(x)], \quad i = 1,2,3 \$

with

$$\Phi_1(x) = a x_1 x_2 - b x_3, \quad \Phi_2(x) = c x_2 x_3 - a x_2, \quad \Phi_3(x) = b x_3 x_1 - c x_2.$$

This is a cubic (third order) stochastic operator for constant $f$, encoding nonlinear interaction effects.

When the speed function $f$ is small (formally, $f/n$ or $f \ll 1$), the model approximates a continuous-time system via a first-order Euler discretization: $x_i(n+1) = x_i(n)[1 + (\Phi_i(x(n)) f(x(n))/n]$ which yields the ODE system

$$\frac{dx_i}{dt} = x_i \Phi_i(x) f(x), \quad i=1,2,3.$$

This correspondence ensures that as the speed of updates becomes small, the discrete-time model closely tracks the corresponding continuous-time dynamics.

3. Parameter Regimes and Dynamical Scenarios

The behavior is highly sensitive to the signs and magnitudes of the interaction parameters $(a, b, c)$:

1. Mixed-sign parameters ($ab < 0, ac < 0$, or $bc < 0$):
   • Non-persistence: All orbits, including those initiated in the interior of the simplex, converge to one of the simplex's vertices. In ecological language, only one species survives, the others are driven extinct.
   • The fixed points in this case correspond to the simplex's vertices $e_1 = (1,0,0)$, $e_2 = (0,1,0)$, $e_3 = (0,0,1)$.
2. All positive parameters ($a > 0, b > 0, c > 0$):
   • Weak persistence, non-ergodicity: The system possesses a unique interior fixed point, denoted $x^{*}$, with all $x_i^{*} > 0$ and determined by explicit functions of $(a,b,c)$.
   • However, almost every trajectory starting in the interior does not converge to $x^{*}$. Instead, its Cesàro means (time-averaged states) exhibit the simplex's vertices as accumulation points. This “cyclic dominance” is analogous to rock–paper–scissors dynamics, with extended dominance phases punctuated by switches.
   • The ergodic hypothesis fails: trajectories do not settle to a unique limiting distribution.
3. All negative parameters ($a < 0, b < 0, c < 0$):
   • Strong persistence: Every orbit started in the interior converges to the unique interior fixed point. All species robustly coexist and extinction is avoided.
   • The Lyapunov function $p(x) = x_1 x_2 x_3$ strictly decreases (except at the fixed point), enforcing global convergence.

These regimes are summarized in the following table:

Parameter Signs	Long-Term Behavior	Fixed Points
Mixed (e.g. ab < 0)	One species survives	Vertices only
All positive	Cyclic dominance,	Unique interior FP,
	non-ergodic	cycles on vertices
All negative	Coexistence (converges)	Unique interior FP

4. Dynamical Properties and Ergodicity

For mixed-sign parameter cases, the destruction of coexistence aligns with standard competitive exclusion. However, the all-positive scenario exhibits long, variable intervals where each species in turn dominates, before the next rises to take over—producing recurrent excursions near the boundaries of the simplex. Even though no species goes extinct, the system’s averages (Cesàro means) do not converge in the classical sense; the time average of the state oscillates indefinitely.

For all-negative parameters, the global Lyapunov function guarantees monotonic decay of the “volume” away from the interior fixed point, leading to robust coexistence irrespective of initial conditions within the interior.

The structure of fixed points and their stability is as follows:

• Vertices: Always fixed points. Attractors in the mixed-sign case.
• Interior fixed point: Exists for all-positive or all-negative parameters, computed using parametrizations such as $X_1 = \sqrt{3 b c^2}$ and similar combinations for $X_2, X_3$. Stability and convergence depend on the sign structure.

5. Comparison to Continuous-Time and Quadratic Stochastic Operators

The inclusion of the speed function $f$ offers a direct link to continuous-time analogues: when $f(x)$ is taken small or $f(x)/n$ for large $n$, the model closely approximates Kolmogorov-type ODE systems customary in theoretical ecology. Thus, the initially discrete-time system can interpolate between fully discrete and approach continuous formulations as desired. In particular, such a design connects with the rich literature on quadratic and cubic stochastic operators, but generalizes previous results by incorporating an adaptive time-step and higher-order interactions.

Though fixed points and invariant sets remain unchanged in the limit of small $f$, the fine details of orbits, especially regarding the presence or absence of ergodicity or chaos, may differ.

6. Biological and Mathematical Implications

The discrete-time three-species model reveals that interaction sign and strength directly shape the asymptotic outcomes:

• Coexistence or exclusion is not purely a function of initial conditions but encoded in the system’s interaction structure,
• Cyclic dominance (non-ergodicity with recurrent dominance phases) provides a mechanism for long-term diversity and dynamic instability,
• Strong persistence with convergence is only guaranteed under sufficiently balanced negative (competitive) interactions.

Mathematically, these models enable the paper of ergodicity, persistence, the breakdown of mean-convergence, and have connections to stochastic operators and their generalizations.

From an applied perspective, the findings indicate that managing interaction structures, rather than only initial abundances, may be crucial for long-term ecosystem stability and resilience to extinctions.

7. Summary

A three-species discrete-time dynamical model on the simplex serves as a rich paradigm for understanding interacting populations. Its properties—determined by three interaction parameters—range from complete extinction of all but one species, to robust coexistence, to complex recurrent “on-off” dominance regimes with no mean convergence. The inclusion of a variable speed function allows discrete-time models to interpolate to continuous-time behavior, preserving mathematically and ecologically relevant fixed points. This framework highlights the deep connections between algebraic properties of nonlinear population maps, the topology of state spaces, and fundamental ecological outcomes such as persistence, extinction, and cyclic dominance (Jamilov et al., 2019).