This chain resembles the diamond-shaped'chain that has been studied before fN,M e.g., fN,M E.G. Noonburg and p. a Abrams, Transient dynamics limit the effectiveness of keystone predation in N, bringing about coexistence Amer.Natu.165:322-335 (2005) In this case the two a,RN1 consumers are different 2RN2 species, so there is no movement between the two consumer strategies
M N 1 N 2 R a 2RN 2 a 1RN 1 f2 N 2 M f1 N 1 M This chain resembles the ‘diamond -shaped’ chain that has been studied before; e.g., E. G. Noonburg and P. A. Abrams, Transient dynamics limit the effectiveness of keystone predation in bringing about coexistence, Amer. Natur. 165:322 -335. (2005). In this case the two consumers are different species, so there is no movement between the two consumer strategies
A simple set of equations for this system is as follows dR R = rRl 1 K/a,RN RN Resource ba,RN,-d,N,-fN,M-m, N,+m,N, Consumer Phenotype 1 d ba rn, -d,,-f2N2M+mn,N-m, N, Consumer Phenotype 2 dM CfIN,M+df2N,,M Predator Parameters ai di, and fi may differ between the two consumer phenotypes
1 a1RN1 a2RN2 K R rR dt dR − − = − 1 1 1 1 1 1 12 1 21 2 1 ba RN d N f N M m N m N dt dN = − − − + 2 2 2 2 2 2 12 1 21 2 2 ba RN d N f N M m N m N dt dN = − − + − cf N M cf N M d M dt dM = 1 1 + 2 2 − m , A simple set of equations for this system is as follows: Parameters ai , di , and fi may differ between the two consumer phenotypes. Resource Consumer Phenotype 1 Consumer Phenotype 2 Predator
There are two equilibrium points for the resource and consumers alone d R1 N12=0M1=0 Kbe R2 0N2 M,=0 Kba However, the equilibrium below cannot exist, as only one species can survive in this model of exploitative competition. The better forager excludes the other. R2 Ka.b
1 0 0 1 2 1 1 1 1 1 1 1 1 1 = = = = − * * , * , * N M Kba d a r N ba d R 0 1 0 2 2 2 2 2 1 2 2 2 2 2 = = = = − * * , * , * M Kba d a r N N ba d R There are two equilibrium points for the resource and consumers alone However, the equilibrium below cannot exist, as only one species can survive in this model of exploitative competition. The better forager excludes the other. 1 1 0 2 2 2 2 2 2 2 2 2 2 1 2 2 2 = = − = = − * * , * , * M Ka b d a r N Ka b d a r N ba d R
But at least one of the tri-trophic chains is assumed exist; i. e the better forager is poorer at evading the predator d R3 I-ad N32=0M;=如aK (rcf -a, dm) R=K a,d N 、ba2K f Assume that (1)exists; that is, that cfINu-dm>0 This provides a path to the full system if consumer 2 can invade; i.e., if ba2R -d2-f2M;=ba2 K1 d2-(2/0k(1%d This means that the poorer forager can now invade, because the predator suppresses the better forager to some extent
1 1 2 1 1 1 1 3 2 3 1 3 1 1 1 3 1 0 f d (rcf a d ) rcf ba K N M cf d N rcf a d R K m * * , * m , * m = = = − − = − 1 2 2 2 2 2 2 4 2 4 1 4 2 2 2 4 1 0 f d (rcf a d ) rcf ba K M cf d N N rcf a d R K m * m * , * , * m = = = − − = − But at least one of the tri-trophic chains is assumed exist; i.e., the better forager is poorer at evading the predator. 1 1 1 − m 0 * cf N , d (1) (2) Assume that (1) exists; that is, that This provides a path to the full system, if consumer 2 can invade; i.e., if 1 1 0 1 1 1 2 2 1 1 1 1 2 3 2 2 3 2 − − − − − − = − d ) rcf a d d ( f / f )( ba K rcf a d ba R d f M ba K * * m m This means that the poorer forager can now invade, because the predator suppresses the better forager to some extent
To obtain this solution we also make the assumption that the consumers are in an Ideal Free Distribution at equilibrium, so that m12N,*=m2 N2 This means that the individuals are distributed among the two strategies such that changing will not improve their fitness there is still switching but it is balanced We can find the interior equilibrium point Rs fd-fd b(a,2-a,f M ba,(2d,-fid f b(,fs)fr ad f2[Kba21-a12)-(fd1-f1d2 c(a21-a12 Kb(afi -a,f2) f,r/ Kb(a2fi-a,f2)-(f2d -f,d2)] c(a2fi-a,f2 Kb(a,fi-a,f2)
b( a f a f ) f d f d R * 1 2 2 1 2 1 1 2 5 − − = ( ) 1 1 1 2 2 1 2 1 1 2 1 1 5 f d b( a f a f ) f d f d f ba M * − − − = 2 2 1 1 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 5 1 Kb( a f a f ) f r[ Kb( a f a f ) ( f d f d )] c( a f a f ) a d N * m , − − − − − − = 2 2 1 1 2 1 2 1 1 2 2 1 1 2 2 1 1 2 1 5 2 Kb( a f a f ) f r[ Kb( a f a f ) ( f d f d )] c( a f a f ) a d N * m , − − − − + − − = To obtain this solution we also make the assumption that the consumers are in an Ideal Free Distribution at equilibrium, so that m12N1* = m21N2*. This means that the individuals are distributed among the two strategies such that changing will not improve their fitness. There is still switching, but it is balanced. We can find the interior equilibrium point