Stability of cycles and survival in a Jungle Game with four species

Abstract

The Jungle Game is used in population dynamics to describe cyclic competition among species that interact via a food chain. The dynamics of the Jungle Game supports a heteroclinic network whose cycles represent coexisting species. The stability of all heteroclinic cycles in the network for the Jungle Game with four species determines that only three species coexist in the long-run, interacting under cyclic dominance as a Rock-Paper-Scissors Game. This is in stark contrast with other interactions involving four species, such as cyclic interaction and intraguild predation. We use the Jungle Game with four species to determine the success of a fourth species invading a population of Rock-Paper-Scissors players.

Figures (8)

• Figure 1: The cyclic relationships of four species: arrows point from winner to loser.
• Figure 2: The relationships of four species in the Jungle game; arrows point from winner to loser. We distinguish three strengths of interaction: directly from $S_i$ to $S_{i+1}$ by a dashed line, from $S_i$ to $S_{i+2}$ by a solid line, and by a dash-dotted line the interaction between the lowest species $S_4$ and the top species $S_1$.
• Figure 3: A $\Delta$-clique $\Delta_{ijk}$. The short connection is $C_{ik}$, the first-long connection is $C_{ij}$ and the second-long connection is $C_{jk}$.
• Figure 4: (a) The relationships of four species in the Jungle game; arrows point from winner to loser. (b) The four species Jungle game network: 2-dimensional connections are indicated by the letter A and solid lines, 1-dimensional connections in a $\Delta$-clique by B and dashed lines, and other 1-dimensional connections by D and dash-dotted lines. The grey arrows signal the existence of a $\Delta$-clique. There are two $\Delta$-cliques in the heteroclinic network of the Jungle game with four species: $\Delta_{321}$ and $\Delta_{432}$.
• Figure 5: A typical time course for \ref{['eq:4-species']}. The lines in dashed grey, solid grey, solid black and dashed black are the coordinates $x_1, x_2, x_3$ and $x_4$ respectively. The parameter values for the simulation are: $c_A=1.2$, $c_B=1$, $c_D=1.1$, $e_A=0.7$, $e_B=0.65$, $e_D=0.72$. The initial condition is $x_1=x_2=x_4=0.1$ and $x_3=1$.

Theorems & Definitions (18)

• Lemma 2.1: Lemma 4.4 from Podvigina_et_al_2020
• Lemma 2.2: Lemma 4.6 from Podvigina_et_al_2020
• Lemma 2.3: Lemma 4.8 from Podvigina_et_al_2020
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Lemma 2.7
• Lemma 2.8
• Theorem 3.1
• proof