Nicholson-Baily+Host-Parasite+system

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=Nicholson-Bailey host-parasite system=

The Nicholson-Bailey model of host-parasite interactions focuses on infection rates by parasites (i.e. their migration) and attempts to understand the evolution of such rates. Moreover for such a system there is debate whether an ODE or MAP is a better descriptor of the system, but that is not the issue here. Here we study this system in space as a Lattice MAP and so study the effects of integrating ecological and evolutionary dynamics as studied in [|Savill et al (1997).] In the model [|parasitoids] kill hosts and migrate with parameter B. When B=1 all hosts get the same amount of parasite according to host density. When B=0 differentiation over hosts is random and when B>1 parasitoids always go to host with highest density. In the model, B is mutated.

Result show both spiral patterns and more chaotic regions (Fig. 1) and it can be shown that there are 3 levels of selection:
 * host / parasite - level
 * spiral wave / chaotic wave - level
 * regions of spiral waves / chaotic waves - level

By studying the life-cycle of spatial patterns in the model it becomes clear that there is an emergence of a [|life-history] of spirals. Basically high B leads to chaotic waves and low B leads to spirals, thus the evolution of B determines what spatial patterns emerge, and this happens in the following way:
 * new spirals are born at the interface between spiral and chaotic wave regions with high B as descended from chaotic waves
 * these spirals rotate fast with active migration and thus increase their domain
 * however when living in a spiral B evolves to lower values and spirals lose domain

So why does this happen? Well, since all individuals in a spiral come from the core, low B has more of a chance to stay in the core (others migrate out!).

It becomes clear that there are various selection pressures especially when inclusive fitness is analysed over different time scales (FIG), i.e. how much fitness individuals have over the generations. When viewed over 50 generations, low B has the least offspring, no matter what spatial pattern individuals are in. Over 300 generations can be seen to have greatest fitness within spiral cores and also dominates over other locations (spiral arms and chaotic waves) and wins over the long term. So does everyone evolve to low B values? Clearly this does not happen, and the explanation is that over the 300 generations descendants from low B ancestors will always evolve to higher B values. From these results we can conclude that
 * there is no fixed [|fitness] which remains the same
 * fitness is not fixed over longer time points
 * at one time point lower fitnesses may be the major source of offspring in the system
 * there are multiple evolutionary time scales in the system
 * "[|good of species]" is obvious not stable against immediate fitness, but the effects of immediate fitness may be long term fitness!

Next: Eco-Evo Predator-Prey model (vd Laan)

 **Savill, N.J., Rohani, P. and P. Hogeweg** (1997) Self-reinforcing spatial patterns enslave evolution in a host parasitoid system. J. theor. Biol. 188, 11-20. [|PDF-file]
 * References**