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13th International Conference on Membrane Computing - MTA Sztaki

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Modelling ecological systems with the calculus of wrapped compartments<br />

for individuals of two species a and b using the following stochastic rewrite rules:<br />

l : a b fa·|α ab|<br />

↦−→<br />

{ a a b if αab > 0<br />

b if α ab < 0<br />

l : a b f b·|α ba |<br />

↦−→<br />

{ a b b if αba > 0<br />

a if α ba < 0<br />

where f i = ri<br />

K i<br />

is obtained from the usual growth rate and carrying capacity. The<br />

α coefficients are put in absolute value to compute the rate of the rule, their signs<br />

affect the right hand part of the rewrite rule.<br />

Example 3. Mutualism has driven the evoluti<strong>on</strong> of much of the biological diversity<br />

we see today, such as flower forms (important to attract mutualistic<br />

pollinators) and co-evoluti<strong>on</strong> between groups of species [45]. We c<strong>on</strong>sider two<br />

different species of pollinators, a and b, and two different species of angiosperms<br />

(flowering plants), c and d. The two pollinators compete between each other, and<br />

so do the angiosperms. Both species of pollinators have a mutualistic relati<strong>on</strong><br />

with both angiosperms, even if a slightly prefers c and b slightly prefers d. For<br />

each of the species involved we c<strong>on</strong>sider the rules for the logistic model and for<br />

each pair of species we c<strong>on</strong>sider the rules for competiti<strong>on</strong> and mutualism. The<br />

parameters used for this model are in Table 1. So, for example, the mutualistic<br />

relati<strong>on</strong>s between a and c are expressed by the following CWC rules<br />

⊤ : a c<br />

ra<br />

Ka<br />

↦−→ ·αac<br />

rc<br />

Kc<br />

a a c ⊤ : a c ↦−→<br />

·αca<br />

a c c<br />

Figure 3 shows a simulati<strong>on</strong> obtained starting from a system with 100 individuals<br />

of species a and b and 20 individuals of species c and d. Note the initially<br />

balanced competiti<strong>on</strong> between pollinators a and b. This random fluctuati<strong>on</strong>s are<br />

resolved by the “l<strong>on</strong>g run” competiti<strong>on</strong> between the angiosperms c and d: when d<br />

predominates over c it starts favouring the pollinator b that now can win its own<br />

competiti<strong>on</strong> with pollinator a. The model is completely symmetrical: in other<br />

runs, a faster casual predominance of a pollinator may lead the evoluti<strong>on</strong> of its<br />

preferred angiosperm. The CWC model describing this example can be found<br />

at: http://www.di.unito.it/~troina/cmc13/compmutu.cwc.<br />

Species (i) r i K i α ai α bi α ci α di<br />

a 0.2 1000 • -1 +0.03 +0.01<br />

b 0.2 1000 -1 • +0.01 +0.03<br />

c 0.0002 200 +0.25 +0.1 • -6<br />

d 0.0002 200 +0.1 +0.25 -6 •<br />

Table 1. Parameters for the model of competiti<strong>on</strong> and mutualism.<br />

3.3 Trophic Networks<br />

A food web is a network mapping different species according to their alimentary<br />

habits. The edges of the network, called trophic links, depict the feeding pathways<br />

395

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