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Dynamics of coupled populations in an open, three-trophic-level system

Model Type
SES Library XPP model

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.  

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Name Initial Value Description
\(A\) - population of the primary producers (the first trophic level)
\(G\) - population of the grazers (the second trophic level)
\(P\) - population of the predators (the third trophic level)
Name Default Value Description
\(I_A\) - Total producer supply rate from external sources (biomass per unit area per unit time)
\(I_G\) - Total grazer supply rate from external sources (grazers per unit area per unit time)
\(I_P\) - Total predator supply rate from external sources (predators per unit area per unit time)
\(r\) - intrinsic growth rate of the primary producers
\(K\) - carrying capacity for the primary producers
\(\sigma_1\) - Grazing coefficient (area per grazer per unit time)
\(\sigma_2\) - Predating coefficient (area per predator per unit time)
\(b_1\) - grazers' handling time per unit of producer biomass
\(b_2\) - predators' handling time per grazer
\(e_{G0}\) - the maximum attainable emigration rate of grazers
\(e_{P0}\) - the maximum attainable emigration rate of predators
\(v\) - sensitivity of grazers' emigration rate to predators' population
\(\alpha_1\) - sensitivity of grazers' emigration rate to producers' population
\(\alpha_2\) - sensitivity of predators' emigration rate to grazers' population
$$\frac{dA}{dt} = I_A + rA(1-\frac{A}{K}) - f_G(A)G$$
$$\frac{dG}{dt} = I_G - f_P(G)P - e_G(A,P)G$$
$$\frac{dP}{dt} = I_P + e_P(G)P$$
$$f_G(A) = \frac{\sigma_1 A}{1+\sigma_1 b_1 A}$$
$$f_P(G) = \frac{\sigma_2 G}{1+\sigma_2 b_2 G}$$
$$e_G(A,P) = e_{G0}(1+vP)exp(-\alpha_1 A)$$
$$e_P(G) = e_{P0}exp(-\alpha_2 G)$$
nisbet_et_al_1997.ode
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