On $ξ^{(s)}$-Quadratic Stochastic Operators on two Dimensional simplex and their behavior

Abstract: A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, it was investigated several classes of QSO. In this paper, we study $ξ^{(s)}$--QSO defined on 2D simplex. We first classify $ξ^{(s)}$--QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.

• The paper classifies ξ^(s)-QSOs into 20 non-conjugate classes using partitioning of trait index sets.
• It analyzes explicit fixed points and periodic cycles, revealing transitions based on the parameter a.
• The study establishes rigorous conditions for convergence, bifurcation, and complex dynamic asymptotics in evolutionary models.

Dynamics and Structure of $ξ^{(s)}$-Quadratic Stochastic Operators on the 2-Simplex

Introduction

Quadratic stochastic operators (QSOs) have long served as a canonical model for the evolution of species distributions in population genetics, capturing the non-linear inheritance mechanisms through their quadratic mapping structure. Despite extensive historical analysis, including their foundational role in discrete-time dynamic systems and their recursive connection to Lotka-Volterra ecological interactions, the classification and long-term dynamic behavior of general QSOs remain unresolved, especially even in low-dimensional settings. The present paper advances this study by systematically developing and analyzing the class of $ξ^{(s)}$-QSOs on the two-dimensional simplex ($S_2$), focusing on a new taxonomy based on partition structures of index sets and an in-depth investigation into the associated operator dynamics.

Classification of $ξ^{(s)}$-QSOs

The main structural contribution of the paper is the exhaustive classification of $ξ^{(s)}$-QSOs on $S_2$, corresponding to partitionings of the coupled index set $P_3$ of parent traits. By careful combinatorial analysis, it is proven that, after accounting for conjugacy under the permutation group action, these operators can be grouped into 20 non-conjugate classes. The authors enumerate 36 parametric forms of $ξ^{(s)}$-QSOs via specific choices of inheritance coefficient vectors $P_{ij}$, which encode the conditional offspring trait distributions depending on parent trait pairings. For each operator, explicit update equations are provided, showing intricate dependencies of next-generation species proportions on quadratic forms of current states and parameter $a \in [0,1]$.

This classification encompasses several well-studied and novel subclasses:

• Volterra-QSOs, where offspring repeat the genotype of one of the parents, corresponding to $P_{ij,k}=0$ for $k \notin \{i,j\}$.
• $l$-Volterra-QSOs, a natural generalization where such a condition is imposed only for the first $l$ traits.
• Permuted $l$-Volterra-QSOs, where operator structure is further modulated by index permutations.
• Strictly singular ($ξ^{(s)}$) classes, where supports of hereditary vectors $(P_{ij})$ are restricted according to the partition and orthogonality conditions.

Of particular note is the rigorous proof that three natural partition classes—corresponding to partitions $\xi_2$, $\xi_3$, and $\xi_4$ of $P_3$—are conjugate and thus yield dynamically equivalent families under suitable coordinate transformation.

Analysis of Operator Dynamics

The paper provides detailed qualitative and quantitative analysis for representative operators from three main classes: $K_1$ (an $l$-Volterra-QSO), $K_4$ (a permuted $l$-Volterra-QSO), and $K_{19}$ (a permuted Volterra-QSO).

Asymptotic Behavior in Class $K_1$

For the $l$-Volterra-QSO $V_{13}$, the fixed point structure, periodic points, and limit sets are precisely characterized as a function of parameter $a$:

• For $a \neq 1/2$: The fixed points are precisely the simplex vertices $e_1, e_2, e_3$.
• For $a = 1/2$: The fixed point set is dramatically enlarged to include the edge $\{x \in S_2 : x_2 = 0\}$ and the line $\{x_1 = x_3\}$.
• Asymptotic Results:
  • If $0 \leq a < 1/2$, all interior points (except those where $x_2 = 0$) converge to $e_2$; otherwise, to $e_3$.
  • If $1/2 < a \leq 1$, orbits converge to $e_1$ except when $x_1 = 0$, in which case $e_2$ is the attractor.
  • For $a = 1/2$, the omega-limit set is either a continuum or a singleton, depending on the initial condition.

Strong claims regarding the monotonicity of the underlying quadratic map $f_a(x) = x^2 + 2a x(1-x)$ (e.g., presence of only trivial fixed points, uniform global convergence determined by $a$) are rigorously established.

Dynamics in Permuted $l$-Volterra and Volterra-QSO Classes (K4, K19)

The authors show that permutation of component indices generates dynamically distinct behaviors, even when the algebraic form is similar. For example, $V_4$ (from $K_4$) possesses for certain parameter values continuum families of fixed points and a nontrivial set of 2-periodic points, leading to oscillatory dynamics along invariant subspaces. The intricate dependence of the asymptotics on analytic properties of the associated quadratic functions is mapped out in detail, with explicit critical values for $a$ controlling transitions between convergence to simplex vertices, cycles, and more complex asymptotics.

For the permuted Volterra-QSO $V_{28}$ (class $K_{19}$), the fixed point structure, order-2 periodics, and regions of attraction are explicitly calculated, leveraging the monotonicity and invariant set structure of associated quadratic or biquadratic mappings. For $a = 1/2$, an exceptional behavior is found where the set of period-2 points fills an entire edge minus one point.

Numerical Interpretation and Summary of Results

Strong results are given regarding the size, location, and stability of fixed points, as well as explicit conditions for periodic cycles and their basins of attraction. The dependence on the parameter $a$ is completely resolved for all operators studied, with all bifurcation values shown in closed form.

Theoretical Implications

This paper's framework for partition-induced classification of QSOs yields a refined understanding of how coupling among traits, encoded in partitioning of $P_m$, restricts or enables complex system evolution. The explicit characterization of fixed and periodic point sets in subclasses such as permuted $l$-Volterra-QSOs, along with precise basin structure, provides a foundation for a deeper understanding of stability and bifurcation phenomena in nonlinear evolutionary systems. The findings emphasize that even in low dimensions, the introduction of partition-based singularity and permutation can induce substantial qualitative shifts in dynamics.

Perspectives and Future Directions

The analytical approach used here can be generalized to higher-dimensional simplices, though the combinatorics and geometry become appreciably more intricate. Future investigations may address more complex partition schemes, connections with invariant measures, Lyapunov structure, and the presence (or absence) of chaos or quasi-periodicity—phenomena which, while not observed in detail here, are conjecturally present in broader QSO classes. Practically, the results inform how genetic systems structurally constrained by mating rules or population subdivisions will evolve—and may serve as blueprints for robust QSO-based models in mathematical genetics, ecology, or evolutionary game theory.

Conclusion

By systematically classifying $ξ^{(s)}$-QSOs on the two-dimensional simplex and conducting an exhaustive analysis of operator dynamics across representative classes, this paper resolves longstanding questions regarding asymptotic behavior and structural taxonomy in quadratic evolutionary dynamics. The explicit links between operator algebraic data, partition combinatorics, and global dynamical behavior set a template for both theoretical expansions and applications in discrete nonlinear models of genetic evolution and beyond (1305.6371).