*How does the model dynamics change if we consider the mortality process in the last model with the uptake of resource limited by the biomass in the first trophic level (*

**N = u**)? Consider even and odd trophic chains and try to make some generaliztions._{0}T_{1}Well, the models' behaviors are interesting and quite divergent. Generalizations are going to be tough. Let's look at the three-trophic-levels model first. With death coefficient (d) = .05, the top predator dies off rapidly and the other two assume a periodicity that is reminiscent of the earlier predator-pray model.

Three levels, d = 0.05 |

If we decrease the mortality to d = .001, we get very different behavior. The first and third tropic levels decline, while the middle level enters a stable oscillation. I was curious how the system would collapse as levels 1 and 3 became extinct, so I extended the model to run to t = 1000, but the dynamics are unchanged. Levels 1 and 3 continue to decline, but are never eliminated, so level 1 continues to act as a conduit of "N" for level two.

Three levels, d = 0.001 |

Turning now to the four-level model, with d = .05

*,*the overall behavior of the system is rather similar to the three-level model with d = .05 (the first set of graphs). There are some quantitative differences, but we see the same oscillations for the two lowest trophic levels (blue and red), and the highest (green) rapidly collapses.

4-levels, d = 0.05 |

4-levels, d = 0.001 |

And indeed, turning u0 up even further (to 1.0) strengthens the inverse relationship between T1 and T3. So perhaps we can conclude that when reourses at the bottom of the trophic web are abundant, they tend to govern the system, and when they are scarce, predation governs the system. That seems intuitive enough.

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