Ecological Archives E088-095-A1

Norio Yamamura. 2007. Conditions under which plants help herbivores and benefit from predators through apparent competition. Ecology 88:1593–1599.


Appendix A. Global stabilities of dynamical systems in Eqs. 1a-c and 9a and b.

I show here the global stabilities of dynamical systems in Eqs. 1a-c and 9a and b. When (4) is satisfied for (1), I define

3
(A.1)

where 6and 9are represented in (2). This function always has positive values for 12, 15 and 18 except for 21 where the value is zero. Using (1) and (2), we can show

2100 .
(A.2)

The first term of the right-hand side is negative because 21001 is an increasing function of y, and the second term is also negative from (4). Therefore, 24decreases monotonically and therefore ( 27), starting from any initial value, approaches 30 where 3000 has the minimum value, which means that 3001 is globally stable. Thus 33 is a Liapunov function.

When the inequality sign is reversed in (4), we can constitute another Liapunov function:

3300
(A.3)

where 44and 47are represented in (3). We can show that

4700
(A.4)

is negative. Therefore, ( 50) approaches 53 where 58 has the minimum value.

Next, I show the global stability of dynamical system (9). When 61, I define

64
(A.5)

where 67and 72are represented in (10). We can show that

75
(A.6)

which is negative. Therefore, ( 78) approaches 7800 where 81 has the minimum value.

When 84, I define

87
(A.7)

where 90. We can show that

93
(A.8)

which is negative. Therefore, ( 96) approaches 101 where 104 has the minimum value.



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