Metapopulations & Patch Simulations -- KVML Model
by Mike Martin

This page generates a plot for a metapopulation of the percentage of the uninhabitable patches; And, for those that are habitable, the proportion of occupied habitable patches and proportion of those habitable patches that are unoccupied are also plotted.   Let p be the proportion of of uninhabitable patches, then 1 - p is the proportion of habitable patches.   Of the habitable patches, consider those occupied, q, and unoccupied, r.   The relationship between p and t is shown in PURPLE below, the relationship between q and t is shown in BLUE, and the relationship for r is given in BROWN. The rate equations for the model are written


Also, note the algebraic condition that p + q + r = 1.   The parameter λ is the rate of patch creation, e is the rate of patch destruction, δ is the rate of population destruction, and β is the rate of propagule reproduction.   Ro (calculated and given above the first graph) is the threshold for persistence.   The phase plane (r vs q) is the second graph and is plotted with stream lines.

Enter in the values of the non-negative parameters λ, β, ε, δ, po, qo, and select the plotting bounds, then press the "Evaluate" button.

λ =  

β =  

e =  

δ =  


po =  

qo =  

ro = 1 - po - qo

0   ≤   t   ≤  

  ≤   q   ≤  

  ≤   r   ≤  

References:

Keymer, J.E., P.A. Marquet, J.X. Velasco-Hernandez, & S.A. Levin. 2000. Extinction thresholds and metapopulation persistence in Dynamic Landscapes. Amer. Natur. 156: 478-494.