Existence of possible boundary equilibria

2 . 4 . Proof By assumption H1, both of the autotrophs are limited by their carrying capacities, and so from Eq. 2.3 x ; i 5 x i b i N − l i − x i = x i g i x i ; i = 1, 2. By the usual comparison theorem Hale, 1969, we have x i B C i = N − l i Let us define, S 1 t = x 1 t + x 2 t. The time derivative along a solution of the system is S : =x 1 g 1 x 1 + x 2 g 2 x 2 − b 1 + b 2 x 1 x 2 − yp 1 x 1 + b 1 x 1 − yp 2 x 2 + b 2 x 2 5 x 1 g 1 x 1 + x 2 g 2 x 2 = − x 1 g 1 0 + x 2 g 2 0 + x 1 g 1 0 + x 2 g 2 + x 1 g 1 x 1 + x 2 g 2 x 2 or, S : 5 −m 1 x 1 + x 2 + C 1 g 1 0 + C 2 g 2 0 + h 1 + h 2 Therefore, S : 1 + m 1 S 1 B k 2 , where k 2 = C 1 g 1 0 + C 2 g 2 0 + h 1 + h 2 Applying a theorem on differential inequalities Birkhoff and Rota, 1982, we obtain 0 5 S 1 5 k 2 m 1 + e − mt S 1 x 1 0, x 2 0. As t “ , 0 B S 1 B k 2 m 1 . Define S 2 t = x 1 t + x 2 t + y. Then S : 2 = x ; 1 + x ; 2 + y ; 2 5 x 1 g 1 x 1 + x 2 g 2 x 2 − o 3 y = − x 1 g 1 0 + x 2 g 2 0 + o 3 y + x 1 g 1 + x 2 g 2 0 + x 1 g 1 x 1 + x 2 g 2 x 2 . Therefore, S : 2 B − m 2 x 1 + x 2 + y + C 1 g 1 + C 2 g 2 0 + h 1 + h 2 , or S : 2 + m 2 S 2 B k 2 and 0 B S 2 B k 2 m 2 as t “ . Hence system 2.3 is dissipative with the asymptotic bound k 2 m 2 . Thus there is a compact neighbourhood B¤R + 3 such that for sufficiently large T = T x ,t , xt B for all t \ T, where xt = {x 1 t, x 2 t , yt} is a solution of 2.3 that initiates in R + 3 . This completes the proof of the lemma.

3. Existence of possible boundary equilibria

In system 2.3, when all species are absent, the trivial equilibrium E 0,0,0 always exists. The existence of other boundary equilibria depends critically on the amount of total nutrient N in the environment. The axial equilibria E 1 N − l 1 , 0, 0 and E 2 N − l 2 , 0, 0 exist provided N \ l 1 and N \ l 2 , respectively. We now investigate the inte- rior equilibrium in the positive x 1 − y plane.Here isoclines are x ; 1 = 0, or y = F 1 x 1 ; Nwhere F 1 x 1 ; N = b 1 N − l 1 − x 1 p 1 x 1 x 1 + b 1 3.1 and y ; =0 is the vertical line x 1 = m 1 , where m 1 is finite. From Eq. 3.1, F 1 N − l 1 ; N = 0 and F 1 0; N = lim F 1 x 1 ;N = b N − l 1 p 1 0 + b 1 \ d F 1 0;N d x 1 = lim x 1 “ 0 + F 1 x 1 ;N = − 2b 1 p 1 0 + b 1 + b 1 N − l 1 p ¦0 2{p 1 0 + b 1 } 2 a possible configuration of the autotroph isoclines is given in Fig. 1. Hence the unique interior equilibrium E 13 m 1 ,0, b 1 N − l 1 − m 1 p 1 m 1 m 1 + b 1 exists in the x 1 − y plane provided N \ l 1 + m 1 3.2 Fig. 1. Æ Ã Ã Ã È H 1 x 1 ,x 2 ,y − b 1 x 1 − p 1 x 1 + b 1 x 1 − x 2 b 2 H 2 x 1 ,x 2 ,y − p 2 x 2 + b 2 x 2 p 1 x 1 y p 2 x 2 y − o + p 1 x 1 + p 2 x 2 Ç Ã Ã Ã É where H 1 x 1 ,x 2 ,y = b 1 N − l 1 − 2b 1 x 1 − b 1 x 2 − yp 1 x 1 + b 1 and H 2 x 1 ,x 2 ,y = b 2 N − l 2 − b 2 x 1 − 2b 2 x 2 − yp 2 x 2 + b 2 . 4.1 Evaluating V at different equilibria, we get their local stability properties. 4 . 1 . Case 4 . 1: N B l 1 B l 2 E o is a sink i.e. E o is saturated. No organism can live in the environment. 4 . 2 . Case 4 . 2: l 1 B N B l 2 E and E 1 exists and no other boundary equi- libria are feasible. E is a hyperbolic saddle point or equivalently non-saturated. It has along each of the y-axis and x 2 -axis a non-empty one dimen- sional local stable manifold. The flows along the x 1 and x 2 axes are away from E and approach E 1 . E 1 is stable or unstable locally along the y-direction according as − o 3 + p 1 N − l 1 B or \ 0, i.e. N B or \ l 1 + m 1 . So E 1 is unstable along y-direction and y always in- creases as its invasion parameter − o 3 + p 1 N − l 1 \ 0 at the axial equilibrium E 1 if N \ l 1 + m 1 . Hence E 1 becomes a saturated equilibrium for l 1 B N B l 1 + m 1 B l 2 4.2 E 1 becomes non-saturated for l 1 B l 1 + m 1 B N B l 2 . 4.3 We now have the following result. 4 . 3 . Theorem 4 . 1 Suppose that l 1 B N B l 1 + m 1 B l 2 . Then any solution of system 2.3 with x i 0 \ 0, y0 \ 0 satisfies Similarly, under H1 E 23 0,m 2 , b 2 N − l 2 − m 2 p 2 m 2 m 2 + b 2 exists in x 2 − y plane provided N \ l 2 + m 2 . 3.3 From Eq. 2.3, it follows that if the herbivore is absent, two autotrophs can not attain E 12 x 1 ,x 2 ,0 and they are in a dominance state. Existence of different axial and planar equi- libria thus depends critically on the total nutrient N, l i and m i ; i = 1,2. It is difficult to determine the interior equi- librium Ex 1 ,x 2 ,y but one can gain informa- tion about whether the two autotrophs and the herbivore can co-exist or not.

4. Stability of equilibria