Robust steady states in ecosystems with symmetries

Figures & data

Figure 1. Graphical representations of structured systems and symmetric structures. (a) Robust structured system corresponding to Equation (2(2) $\begin{array}{rl}{c}_1& =f_1(x_3,x_4,x_5)\\ c_2& =f_2(x_3,x_4,x_5)\\ c_3& =f_3(x_1,x_5)\\ c_4& =f_4(x_1,x_5)\\ c_5& =f_5(x_1,x_2,x_5).\end{array}$(2) ), or equivalently, relation (6(6) $F_{12}=F_{21}=F_{34}=F_{43}=F_{32}=F_{42}=F_{53}=F_{54}=F_{11}=F_{22}=F_{33}=F_{44}=0,$(6) ). (b) Removing two edges results in a fragile system corresponding to relations (6(6) $F_{12}=F_{21}=F_{34}=F_{43}=F_{32}=F_{42}=F_{53}=F_{54}=F_{11}=F_{22}=F_{33}=F_{44}=0,$(6) ) and (7(7) $F_{14}=F_{24}=0.$(7) ). (c) A symmetric structure, as discussed in Section 4. Blue pairs and green pairs are matched Jacobian entries.

Figure 2. The directed graph that corresponds to the Competitive Exclusion Principle. A model of two species that depend only on a single third species has only nonrobust solutions.

Figure 3. Graphical representations of structured systems and symmetric structures. (a) Robust structured system corresponding to relations (9(9) $\begin{array}{rl}{\dot{x}}_1& =-a_1+g_1(x_2,x_3,x_4,x_5)\\ {\dot{x}}_2& =-a_2+g_2(x_2,x_3,x_4,x_5)\\ {\dot{x}}_3& =-a_3+g_3(x_2,x_3,x_4,x_5)\\ {\dot{x}}_4& =-a_4+g_4(x_1,x_2,x_3,x_5)\\ {\dot{x}}_5& =-a_5+g_5(x_1,x_2,x_3,x_4)\end{array}$(9) ). (b) Robust system in (a) is changed to fragile symmetric system when some edges are identified, corresponding to system (10(10) $\begin{array}{rl}{\dot{x}}_1& =-a_1+(x_3+x_4+x_5)g_1\left(x_2\right)\\ {\dot{x}}_2& =-a_2+(x_3+x_4+x_5)g_2\left(x_2\right)\\ {\dot{x}}_3& =-a_3+(x_3+x_4+x_5)g_3\left(x_2\right)\\ {\dot{x}}_4& =-a_4+g_4(x_1,x_2,x_3,x_5)\\ {\dot{x}}_5& =-a_5+g_5(x_1,x_2,x_3,x_4)\end{array}$(10) ). Blue, green and red triples are matched Jacobian entries. (c) With an added edge from node 1 to node 2, robustness is restored in the symmetric structure.