##### Model Description

This model explicitly incorporates the immigration and emigration processes in the context of three-trophic-level systems (e.g., a stream reach), and reveals how the coupled populations will change in response to the primary-productivity gradients (e.g., r, K). In contrast, traditional simple food-chain models study mostly two-trophic-level system which can be open or close. More interstingly, though not reflected here, there are two contrasting models presented in the paper: the discrete-grazer model and individual-based model. The extension from ODE, to pseudo-ABM, to ABM, and how the former becomes the special case of the latter, are very insightful. Readers are encouraged to take a glance at the original paper.

$\frac{dA}{dt} = I_A + rA(1-\frac{A}{K}) - f_G(A)G$ |

The model characterizes the processes underlying the population dynamics of producers. For the right hand side, the first term is the external immigration at a constant rate per unit area; the second term represents the internal logistic growth, where r represents the maximum attainable growth rate, attained only when A is close to zero, and K is the carrying capacity, set by light or nutrients depending on the system under study; the third term is the functional response of producers to grazing. In contrast, the population dynamics of grazers and predators are characterized as following: |

$\frac{dG}{dt} = I_G - f_P(G)P - e_G(A,P)G$ |

for grazers, where the first term in the right hand side again represents the external immigration at a constant rate per unit area; the second term is the functional response of grazers to predating; the last term is the per capita emigration rate of grazers determined by the population density of producers and predators. |

$\frac{dP}{dt} = I_P + e_P(G)P$ |

for predators, where the first term in the right hand side again represents the external predator immigration at a constant rate per unit area; the second term is the per capita emigration rate of predators determined by the population density of grazers. |

$f_G(A) = \frac{\sigma_1 A}{1+\sigma_1 b_1 A}$ |

Above specifies a type II functional response of producers to grazing. Similarly, we have |

$f_P(G) = \frac{\sigma_2 G}{1+\sigma_2 b_2 G}$ |

for a type II functional response of grazers to predating. Besides, assume the following functions for the emigration rate of grazers and predators: |

$e_G(A,P) = e_{G0}(1+vP)exp(-\alpha_1 A)$ |

which decreases with increasing producers, and increases with increasing predators. |

$e_P(G) = e_{P0}exp(-\alpha_2 G)$ |

which is only decreases with increasing grazers. |

# xppaut file for the ODE model in Nisbet et al. 1997 # by Bingbing Zhou, 10/09/2016 par Ia=1,Ig=1,Ip=1,r=0.5,K=1000000,sigma1=0.5,sigma2=0.5,b1=0.5,b2=0.5 par eg0=1,ep0=1,v=0.5,alpha1=0.1,alpha2=0.1 dA/dt = Ia + r*A*(1-A/K) - G*sigma1*A/(1+sigma1*b1*A) dG/dt = Ig - eg0*(1+v*P)*exp(-alpha1*A)*G - P*sigma2*G/(1+sigma2*b2*G) dP/dt = Ip - ep0*exp(-alpha2*G) done

Zhou B, Arizona State University.

Primary-productivity gradients and short-term population dynamics in open systems. Ecological Monographs. 67(4):19.

. 1997.