PHY198S

Physics at the Cutting Edge (2020-21) -- PHY198S

Week 5 preparation

Modelling biological interactions

Suppose that we have $N$ different species, and that they have various types of biological interactions with each other -- like predator-prey, mutually beneficial, asymmetrically toxic, etc. The generalized Lotka-Volterra (gLV) equations describe a specific kind of simple model for how their abundances change over time.

The gLV model rests upon a set of simplifying assumptions, including:-

  1. the rate of population growth of a given species is proportional to its population size;
  2. prey species never go hungry;
  3. all species have a ginormous appetite;
  4. the environment around all these species taken together doesn't change;
  5. genetic adaptation can be ignored;
  6. species don't have any culture.

Obviously, these assumptions are overly simplistic compared to the real world -- for example, the environment cannot be considered to be an infinite resource bank and an infinite trash can. But the gLV model provides a basic way to get started on understanding the dynamics of the combined system of interacting species.

Let $x_i(t)$ be the abundance of species $i$ at time $t$. Then the rate of change in the abundance for species $i$ with time will have two types of contributions. The first type is from population growth through reproduction. Quantitatively, it is given by the number of organisms of the same species, times its growth coefficient $b_i$, which is positive because parents don't have negative numbers of kids. The second type of contribution comes from interactions. Quantitatively, it is given by the abundance of the $i$th species, times the strength of its interaction with a different $j$th species, times the abundance of that other species, summed over all species $j$. Each entry of the interaction strength matrix $a_{ij}$ can be positive or negative (or zero), depending on the type of relationship the $i$th and $j$th species have with each other. In total, we have $$ {\frac{dx_i}{dt}} = x_i(t) \left( b_i + \sum_{j=1}^N a_{ij} x_j(t) \right) \,. $$ If there were zero interactions, the solutions for the abundances would simply grow exponentially with time: $x_i(t)=x_{i,0} e^{b_i (t-t_0)}$. Because there actually are interactions, encoded in the interaction matrix $a_{ij}$, these $N$ equations are coupled and nonlinear. The nonlinearity is why interesting behaviours can happen, like waxing and waning, or population collapse.

Wikipedia reading

Please have a good read of these two pages:

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