fN,M fN,M E.G. Noonburg and p. a Abrams (2005) studied the stability of the diamond shaped food web N, They found that loca stability of the interior equilibrium point was pe ossible a,RN1 2RN2
M N 1 N 2 R a 2RN 2 a 1RN 1 f2 N 2 M f1 N 1 M E. G. Noonburg and P. A. Abrams (2005) studied the stability of the diamond - shaped food web. They found that local stability of the interior equilibrium point was possible
From Noonburg and abrams American Naturalist 2005 However, they found that the system exhibits slowly 亏02 damped extreme 号0.15 fluctuations when the second consumer is 之之 introduced at small values 2004006008001000 This would likely lead to extinction of one or more of the species 03 0.1 2004006008001000 Figure 2: Dynamics of invasion of the three-species community with prey I by an edible prey (N, l. In A, a solid line represents N and a dashed line represents N, In B, a dashed line represents P, and a soid line represents R, Initial conditions for the three resident species are their equilibrium densities in the absence of N R-0.85, N, =0.05. P=0.075. Invasion begins at I a g wim N, E D D00l, Parameter values are as follow: r=l,K=1,4=3,6=6. b=l/3. b= l3, d=0.l,a=048.=104=24=10=10D=905 The fout-speries equibrium is K=0.2535, N =0.%,N
However, they found that the system exhibits slowly damped extreme fluctuations, when the second consumer is introduced at small values. This would likely lead to extinction of one or more of the species. From Noonburg and Abrams , American Naturalist 2005
The question then is, what is the stability behavior of the continuous switching among the phenotypes occur? ich analogous model for two consumer phenotypes in wl Analysis of the eigenvalues from the matrix R3--a1R3-a2R30 K ba, M fN3.1=0 baN -f2 0 shows that local sta bility is possible
The question then is, what is the stability behavior of the analogous model for two consumer phenotypes in which continuous switching among the phenotypes occurs? Analysis of the eigenvalues from the matrix 0 0 0 1 3 2 3 2 3 2 12 21 2 3 2 1 3 1 12 21 1 3 1 3 1 3 2 3 = − − − − − − − − − − * * * , * , * , * , * * * cf M cf M ba N m m f N ba N m m f N R a R a R K r shows that local stability is possible
The eigenvalues can be further studied as a function of the movement rate m12r with m21 given by 0.005 n12N1*/N 0 For m22 small, -<-005 0.01 0.015 dynamics is -0.005 dominated by a-0.01 complex conjugate eigenvalues with small 三-0.015 negative real part -0.02 As m,? is increased, 0.02 the absolute value of Movement Rate, m12 the real part increases and then the solution bifurcates to two real eigenvalues
The eigenvalues can be further studied as a function of the movement rate m12, with m21 given by m12 N1*/N2 *. For m12 small, dynamics is dominated by complex conjugate eigenvalues with small negative real part. As m12 is increased, the absolute value of the real part increases and then the solution bifurcates to two real eigenvalues
Simulations For small m,a in the two consumer phenotype system, slowly damped large oscillationsoccur. as n Noonburgand N 2 Abrams (2005) 30 model of a diamond z, !"! shaped web. 100 20004000600D 000100001200014000160
R* N2 * N1 * For small m12 in the two consumer phenotype system, slowly damped large oscillations occur, as in Noonburg and Abrams (2005) model of a diamondshaped web. Simulations