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Coexistence coalitions in propagule disperser quasi-communities

Leonardo Aguirre, José A. Capitán, David Alonso

Abstract

Many natural ecosystems harbor large numbers of coexisting species competing for far fewer distinct resources, in apparent defiance of the competitive exclusion principle. Various mechanisms have been proposed to explain this apparent paradox, among the most prominent being competition--colonization trade-offs, environmental heterogeneity, and ecological neutrality. We develop a unified stochastic model class that combines all three coexistence narratives in the context of propagule disperser communities and show that this setting encompasses several important classical models. We then prove a general theorem on coexistence at macroscopic equilibria and provide an algorithm that determines equilibrium coalitions solely from readily available matrix spectra, thereby bypassing the costly computation of exact equilibrium states. Using illustrative examples, we demonstrate the potential of this approach for quantifying the relative merits of different coexistence narratives and for studying their synergistic effects.

Coexistence coalitions in propagule disperser quasi-communities

Abstract

Many natural ecosystems harbor large numbers of coexisting species competing for far fewer distinct resources, in apparent defiance of the competitive exclusion principle. Various mechanisms have been proposed to explain this apparent paradox, among the most prominent being competition--colonization trade-offs, environmental heterogeneity, and ecological neutrality. We develop a unified stochastic model class that combines all three coexistence narratives in the context of propagule disperser communities and show that this setting encompasses several important classical models. We then prove a general theorem on coexistence at macroscopic equilibria and provide an algorithm that determines equilibrium coalitions solely from readily available matrix spectra, thereby bypassing the costly computation of exact equilibrium states. Using illustrative examples, we demonstrate the potential of this approach for quantifying the relative merits of different coexistence narratives and for studying their synergistic effects.
Paper Structure (32 sections, 11 theorems, 98 equations, 4 figures, 1 table)

This paper contains 32 sections, 11 theorems, 98 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Classical coexistence mechanisms
  3. Displacement competition
  4. Environmental heterogeneity
  5. Ecological neutrality
  6. Propagule disperser quasi-communities
  7. State space
  8. Stochastic kinetics
  9. Macroscopic approximation
  10. Classical cases
  11. Equilibrium coexistence
  12. The trait-drift model
  13. The general case
  14. Discussion
  15. Mathematical terminology and notations
  16. ...and 17 more sections

Key Result

Proposition 1

The macroscopic vector field of the continuous-time Markov chain defined by the kinetic events $(r_k, V_k), k\in\mathfrak{K}$, takes the form where $C_1,\ldots, C_5$ are diagonal matrices containing the propensity coefficients, $D$ and $\tilde{D}$ are matrices containing the dispersal/mutation and migration probabilities, $P_0$ is a diagonal matrix containing the basal establishment probabilities

Figures (4)

  • Figure 1: The limit cycle attractor from Example \ref{['exmp:trait-drift_counterexample']} with sessile coordinates on the left and propagule coordinates on the right. The direction of motion along the attractor is indicated by arrow tips and the red dot locates the associated equilibrium.
  • Figure 2: Illustration of Example \ref{['exmp:trait-drift']}. Top: Equilibrium abundances of sessile trait-types. Bottom left: Sessile mortality coefficients $c_{1,i}$ and dispersal coefficients $c_{3,i}$ with $i=(1,a,1)$ as a function of trait-type $a=0,\ldots,10$. Bottom right: Phase diagram for deformed nonlinearity and spread of trait values (peaks/valleys including those at the boundary).
  • Figure 3: Illustration of the setup in Example \ref{['exmp:general']} showing the three-by-three grid of macrosites with diagonal resource availability gradient and dispersal connections indicated.
  • Figure 4: Coalition phases for Example \ref{['exmp:general']} in heterogeneity superparameter space for six values of displacement strength (values in main text). Environmental heterogeneity is indicated on the horizontal $s_1$-axis and trait heterogeneity on the vertical $s_2$-axis.

Theorems & Definitions (23)

  • Proposition 1: macroscopic vector field
  • proof : Proof (sketch, details in Appendix \ref{['app:macroscopic_vf']})
  • Corollary 1
  • proof : Proof (sketch)
  • Proposition 2: equilibrium condition
  • proof : Proof (sketch, details in Appendix \ref{['app:equilibrium_condition']})
  • Proposition 3: quasi-static limit
  • proof : Proof (sketch, details in Appendix \ref{['app:quasi-static']})
  • Proposition 4
  • proof : Proof (sketch, details in Appendix \ref{['app:trait-drift_proof']})
  • ...and 13 more