Dynamics of Two Species with Density-Dependent Interactions in a Mutualistic Context

TL;DR

A general deterministic model of mutualism involving two populations, assuming that mutualism may involve both costs and benefits for the interacting individuals, leading to density-dependent effects on the dynamics of the two species.

Abstract

Mutualistic interactions, where individuals from different species can benefit from each other, are widespread across ecosystems. This study develops a general deterministic model of mutualism involving two populations, assuming that mutualism may involve both costs and benefits for the interacting individuals, leading to density-dependent effects on the dynamics of the two species. This framework aims at generalizing pre-existing models, by allowing the ecological interactions to transition from mutualistic to parasitic when the respective densities of interacting species change. Through ordinary differential equations and phase portrait analysis, we derive general principles governing these systems, identifying sufficient conditions for the emergence of certain dynamic behaviors. In particular, we show that limit cycles can arise when interactions include parasitic phases but are absent in strictly mutualistic regimes. This framework provides a general approach for characterizing the population dynamics of interacting species and highlights the effect of the transitions from mutualism to parasitism due to density dependence.

Figures (10)

• Figure 1: Illustration of density-dependent effects of mutualism (Definition \ref{['def:mut']}) through the signs of cross partial derivatives. In both situations, the interaction is classified as mutualistic according to Definition \ref{['def:mut']}, even though the local effects of one species on the other vary. Figure \ref{['fig:schema_mut']}: $\frac{\partial f}{\partial y}>0$ everywhere, meaning that species $y$ always has a positive effect on the growth of species $x$. The hatched region indicates where $\frac{\partial g}{\partial x}>0$, so that species $x$ also benefits species $y$ locally. This region represents a strictly mutualistic zone. Outside this zone, the effect of species $x$ on $y$ is negative, yet the system still qualifies as mutualistic under Definition \ref{['def:mut']}. Figure \ref{['fig:schema_para']}: The cross partial derivatives $\frac{\partial f}{\partial y}$ and $\frac{\partial g}{\partial x}$ vary and are positive in distinct, non-overlapping regions: the hatched (resp. dashed) region corresponds to $\frac{\partial g}{\partial x}>0$ (resp. $\frac{\partial f}{\partial y}>0$). Outside these regions, the interaction is parasitic for both species.
• Figure 2: Two possible local configurations of the phase portrait near an equilibrium point, corresponding to the two admissible clockwise sign changes of the vector field across adjacent regions. Illustration of Proof of Theorem \ref{['thm:equilibrium']}. The curve $\Gamma_g$ is shown in blue, and $\Gamma_f$ in orange. Note: A single arrow is used to represent the direction of the vector field within each region for simplicity. For instance, in the $\left(+,+\right)$ region, the direction of the vector field $\left(\nearrow\right)$ may vary from horizontal $\left(\rightarrow\right)$ (excluded) to vertical $\left(\uparrow\right)$ (excluded), as long as the signs remain strictly positive.
• Figure 3: Example of a mutualism model exhibiting different types of equilibrium points. Phase portrait of system \ref{['eq:ex+1']} for the parameter values: $a_1=6$, $b_1=2$, $c_1=1$, $d_1=1$, $e_1=1$ and $a_2=5$, $b_2=4$, $c_2=1$, $d_2=1/4$, $e_2=1$. The curve $\Gamma_g$ is shown in blue, and $\Gamma_f$ in orange. White points represent repulsive points, black points represent attractive points and gray points represent saddle points. The figure illustrates a mutualistic system with multiple types of equilibrium points.
• Figure 4: Phase portraits of four differents models of mutualism. Each model highlights different aspects of mutualism while satisfying the assumptions \ref{['hyp:one']} to \ref{['hyp:four']} of system \ref{['eq:init']} and exhibiting an alternation of equilibrium point indices along the isoclines. The curve $\Gamma_g$ is shown in blue, and $\Gamma_f$ in orange. White points represent repulsive points, black points represent attractive points and gray points represent saddle points. (\ref{['fig:Zhang']}) Parameters for zhang2003mutualism model: $c_1 = 7$, $c_2 = 8$, $a_1 = 0.7$, $a_2 = 0.5$, $b_1 = 4.5$, $b_2 = 3$, $r_1 = 7$, $r_2 = 4$. (\ref{['fig:Neuhauser']}) For neuhauser2004mutualism model: $r_1 = 1.4$, $r_2 = 1$, $K_1 = 2$, $K_2 = 5$, $\gamma_{12} = 4$, $\alpha_{12} = 1.5$, $a = 0.2$. (\ref{['fig:Graves']}) For graves2006bifurcation model: $r_{10} = -2$, $r_{20} = 0.5$, $r_{11} = 4$, $r_{21} = 5$, $k_1 = 0.2$, $k_2 = 0.2$, $a_1 = 0.35$, $a_2 = 0.5$. (\ref{['fig:Holland']}) For holland2010 model: $r_1 = 2$, $r_2 = 3$, $c_1 = 3$, $c_2 = 1$, $a_{12} = 1$, $a_{21} = 2.6$, $h_1 = 1$, $h_2 = 1$, $q_1 = 1$, $q_2 = 2.7$, $\beta_{12} = 1$, $\beta_{21} = 1.5$, $e_1 = 0.5$, $e_2 = 3$, $s_1 = 0.7$, $s_2 = 0.5$.
• Figure 7: Phase portrait leading to a limit cycle. Simulation of system \ref{['eq:ex_cycle']} with parameters: $a_1=13.69$, $b_1=0.25$, $c_1=4$, $d_1=0.36$ and $a_2=9$, $b_2=0.04$, $c_2=16$, $d_2=1$. The curve $\Gamma_g$ is shown in blue, and $\Gamma_f$ in orange. The white point represents a repulsive point and gray points represent saddle points. The purple trajectory starts at a population density of 2 for species $x$ and 6 for species $y$. It converges toward the limit cycle from the outside, moving away from the extinction states, either of both species or the extinction of $y$. The gray trajectory starts at densities of 7 for species $x$ and 10 for species $y$, and converges toward the limit cycle from the inside, moving away from the interior equilibrium point.

Theorems & Definitions (27)

• Definition 2.1
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Example 3.4
• Proposition 3.5
• Corollary 3.6
• Remark 3.7
• Corollary 3.8
• Corollary 3.9