Invasion fitness in moment-closure treatments (part 2) in The Azimuth Project

+-- {: .standout}
This page is a [[Blog articles in progress|blog article in progress]], written by [[Blake Stacey]].  To discuss as it is being written, go to the [Azimuth Forum](http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=861).
=--

_Continued from [[Invasion fitness in moment-closure treatments|Part 1]]._

### Example 1: Birth, Death, Movement ###

In the previous instalment, we introduced the idea of _pair approximation,_ by which we try to understand a system by tracking the joint probability distributions for pairs of its pieces.  Now, we'll look at this machinery in more detail by focusing on a specific example.  The ecosystem which we shall study will contain one species living on a regular lattice, and the individual organisms of that species can move about, give birth and die.  That is, our pair dynamics will include three processes, each occurring stochastically with its own characteristic rate: _movement_ or _migration,_ _birth_ and _death._  We follow van Baalen (in [Dieckmann _et al._ (2000)](#Dieckmann2000), chapter 19).

(Petri net pictures will go here.)

We write $z$ for the "coordination number" of the lattice.  Birth:
$$
R 0 \rightarrow R R,
$$
with rate $b/z$.

Death:
$$
R a \rightarrow 0 a,
$$
with rate $d/z$.

Movement or migration:
$$
R 0 \rightarrow 0 R,
$$
with rate $m/z$.

$$ 
\begin{array}{rcl}
\frac{d p_{R 0}}{d t} & = & -p_{R 0}[b/z + d + (z - 1)q_{0|R0}m/z + (z - 1)q_{R|0R}(b + m)/z] \\
& & + p_{00} (z - 1) q_{R|00} (b + m)/z \\
& & + p_{R R} [d + (z - 1) q_{0|R R} m/z].
\end{array}
$$

$$ 
\frac{d p_{0 0}}{d t} = -p_{00} 2(z - 1) q_{R|00} (b + m)/z + p_{R 0} 2 [d + (z - 1) q_{0|R 0} m/z].
$$

$$ 
\frac{d p_{R R}}{d t} = p_{R 0} 2[b/z + (z - 1)q_{R|0 R} (b + m)/z] - p_{R R} 2[d + (z - 1) q_{0|R R} m/z].
$$

$$
p_i = \sum_j p_{i j}.
$$

$$
\frac{d p_R}{d t} = (b q_{0|R} - d) p_R.
$$
If we ignore spatial structure altogether, we can say that
$$
q_{0|R} = p_0,
$$
which by normalization of probability means
$$
q_{0|R} = 1 - p_R.
$$
So,
$$
\frac{d p_R}{d t} = (b(1 - p_R) - d) p_R.
$$
This should look familiar: it's a [[logistic equation]] for population growth, with growth rate $b - d$ and equilibrium population $1 - d/b$.

It's worth pausing a moment here and using this result to touch on a more general concern.  Often, a logistic-growth model is presented with the growth rate and the equilibrium population size as its parameters.  When we see the model in that form, we naturally start thinking of those parameters as _independently variable_ quantities.  We imagine that a mutation or a change in the environmental conditions could change one without affecting the other.  However, if the growth rate and the equilibrium population size are both functions of other parameters taken together, then the changes _which are biologically reasonable to consider_ will likely affect both of them.  To understand which quantities we should treat as independent, we need to spend time looking at how the numbers which apply to population-scale phenomena arise from the smaller-scale physiological and ecological goings-on ([Fox 2011](#Fox2011)).

_Continued in [[Invasion fitness in moment-closure treatments (part 3)|Part 3]]._

### References ###

* H. Matsuda, N. Ogita, A. Sasaki, and K. Sato (1992), "Statistical mechanics of population", _Progress of Theoretical Physics_ **88,** 6: 1035--49 ([web](http://ptp.ipap.jp/link?PTP/88/1035/)).
{#Matsuda1992}
* U. Dieckmann, R. Law, and J. A. J. Metz, eds., _The Geometry of Ecological Interactions: Simplifying Spatial Complexity._ Cambridge University Press, 2000.
{#Dieckmann2000}
* T. Gross, C. J. Dommar D'Lima and B. Blasius (2006), "Epidemic dynamics on an adaptive network", _Physical Review Letters_ **96,** 20: 208701 ([web](http://prl.aps.org/abstract/PRL/v96/i20/e208701)). [arXiv:q-bio/0512037](http://arxiv.org/abs/q-bio/0512037).
{#Gross2006}
* A.-L. Do and T. Gross (2009), "Contact processes and moment closure on adaptive networks", in T. Gross and H. Sayama, eds., _Adaptive Networks: Theory, Models and Applications._ Springer.
{#Do2009}
* B. Allen (2010), _Studies in the Mathematics of Evolution and Biodiversity._ PhD thesis, Boston University ([web](http://proquest.umi.com/pqdweb?did=2071736811&Fmt=2&clientId=5482&RQT=309&cfc=1)).
{#Allen2010}
* J. A. Damore and J. Gore (2012), "Understanding microbial cooperation". _Journal of Theoretical Biology_ **299:** 31--41, DOI:10.1016/j.jtbi.2011.03.008 ([pdf](http://gorelab.homestead.com/Papers/UnderstandingMicrobialCooperation.pdf)).
{#Damore2012}
* J. Fox (2011), "[Zombie ideas in ecology: _r_ and _K_ selection](http://oikosjournal.wordpress.com/2011/06/29/zombie-ideas-in-ecology-r-and-k-selection/)" _Oikos_ blog entry.
{#Fox2011}
* B. Simon, J. A. Fletcher and M. Doebeli (2012), "Hamilton's rule in multi-level selection models" _Journal of Theoretical Biology_ **299:** 55--63 [PMID:21820447](http://www.ncbi.nlm.nih.gov/pubmed/21820447).
{#Simon2012}
* B. C. Stacey, A. Gros and Y. Bar-Yam (2011), "Beyond the mean field in host-pathogen spatial ecology", [arXiv:1110.3845 \[nlin.AO\]](http://arxiv.org/abs/1110.3845).
{#Stacey2011}
* J. D. Van Dyken, T. A. Linksvayer and M. J. Wade (2011), "Kin Selection--Mutation Balance: A Model for the Origin, Maintenance, and Consequences of Social Cheating" _The American Naturalist_ **177,** 3: 288--300.  [JSTOR:10.1086/658365](http://www.jstor.org/stable/10.1086/658365) ([pdf](http://openwetware.org/images/0/05/Van_Dyken_et_al_2011_Kin_selection-mutation_balance_Am_Nat.pdf)). 
{#VanDyken2011}

category: blog