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https://github.com/gottacatchenall/ms_dispersal_diffusion_approximation

When can we approximate dispersal with diffusion in ecology?
https://github.com/gottacatchenall/ms_dispersal_diffusion_approximation

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When can we approximate dispersal with diffusion in ecology?

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README 

        
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# Introduction



Human activity is changing the face of Earth, leaving landscapes that

are fragmented and patchy.



It us well understood that landscape structure influences ecosystem

processes [@cite]. Understanding how landscape structure affects

ecological processes remains a fundamental goal of ecological

research.



Landscape connectivity can mitigate the negative effects of habitat

loss on ecosystem functioning, through corridors [@Resasco2019MetDec].



 As a result understanding how habitat structure effects the movement

and dispersal of organisms, and how this scales up to explain the

abundance and distribution of species across space, is a primary aim

of landscape ecology. Models in landscape ecology---analytic,

computational, and statistical--- have long used diffusion to

approximate model how organisms move or disperse between habitat

patches [@Okubo2001DifEco; @Hastings1978GloSta].



What does it mean that model uses diffusion? The way in which

organisms move from one habitat patch to another, via active or

passive dispersal, is inherently stochastic. Diffusion approximates

this stochastic process by assuming the that stochastic process of

movement of organisms between two locations is equal to its expected

value at every time point---ignoring any temporal variation in

dispersal. However, here we show that in some cases this assumption

creates artificially synchronized dynamics across space.



Why is it important we understand when dispersal is a valid

approximation of dispersal? In order to design landscapes that

mitigate biodiversity loss and its effects [@Albert2017AppNet], we

need models to understand how landscape structure affects ecological

processes. Understanding when dispersal is well-approximated by

diffusion, and when it isn't, is important because diffusion models

are much less computationally expensive.



We do this by using a simulation model with two parts: 1) a spatial

graph model of both stochastic dispersal and diffusion, and 2) a

Ricker model of local population dynamics. We then show that there are

two regimes: one under which diffusion creates highly synchronized

dynamics where stochastic dispersal doesn't, and one under which

diffusion and stochastic dispersal produce similar distributions of

synchrony. We show that the boundaries between these regimes is caused

by both the modularity of the dispersal network and demographic

parameters. We show that what distinguishes these regimes is whether

the primary source of variation in population dynamics is either

dispersal or demography.



![TODO Caption](./figures/synchrony_example.png){#fig:example}



# Methods



Here, we present a model of metapopulation dynamics on spatial graphs.

This model contains three parts: a model of landscape connectivity, a

model of local population dynamics, and a model of dispersal. We use

this model to simulate time-series of metapopulation abundances using

both diffusion and stochastic models of dispersal, and then measure

the synchrony of population dynamics between populations. By comparing

the synchrony created by stochastic dispersal and diffusion models, we

show there are two distinct regimes: a regime where diffusion well

approximates stochastic dispersal, and a regime where it does not.



## Landscape connectivity model



Spatial graphs have long been used to model a system of habitat

patches (nodes) connected by dispersal (edges, which combined form a

landscape [@Dale2010GraSpa; @Minor2008GraFra; @Urban2001LanCon].



// have to define connectivity



To describe how the edges of this spatial graph describe dispersal,



we model landscape connectivity as a combination of two different

factors: the probability than any individual migrates during its

lifetime, $m$, and the conditional distribution over spatial nodes of

where an individual goes ($j \in L$), given both that it migrates $m$

and where it started ($i \in L$), which we call the dispersal

potential and denote



$$\Phi_{ij} =  P(i \to j | m)$$



The dispersal potential can be modeled several ways. In empirical

systems, the relative cost of movement from one point to another is

often estimated with resistance surfaces [spear_use_2010]. Here we

model the dispersal potential using isolation-by-distance (IBD), which

assumes the relative probability of dispersal from location $i$ to

location $j$ is inversely proportional to the distance between them,

$d_{ij}$, and the strength of this IBD relationship, $\alpha$, which

is treated as an intrinsic value of a species dispersal capacity. The

form of the IBD relationship (historically called the dispersal

kernel) we consider an exponential with decay-strength $\alpha$ and a

cutoff value $\epsilon$ [@Grilli2015MetPer; @Hanski1994PraMod].



$$f(d_{ij}, \alpha, \epsilon) =  \begin{cases} e^{-\alpha d_{ij}}

\quad\quad\quad &\text{if}\quad e^{-\alpha d_{ij}} > \epsilon \ \

\text{and } i \neq j \\   0 &\text{else} \end{cases}$$



Then, to construct a dispersal potential $\Phi_{ij}$ with a kernel

$f(d_{ij}, \alpha)$,  we normalize:



$$\Phi_{ij} = \frac{f(d_{ij}, \alpha, \epsilon)}{\sum_k

f(d_{ik},\alpha, \epsilon)}$$



Note that the sum of each row of $\Phi$, forms a probability

distribution, i.e. $\sum_j \Phi_{ij} = 1 \ \ \forall i$, meaning the

probability that an individual leaves its original population given

that it migrates is 1. In some cases, for a given location $i$, the

dispersal kernel $f(d_{ij}, \alpha, \epsilon)$ could be $0$ for all

$j$, in which case $\Phi_{ii}$ is set to $1$ to enforce this

condition. In all other cases, $\Phi_{ii}=0$. Also note that if

$\alpha=0$, the dispersal potential is a uniform distribution over

other locations. In Figure \ref{fig:mp}, we can see the same set of

points plotted spatial graphs plotted representing the same set of

populations across differing values of isolation-by-distance strength,

$\alpha$.



## Local population dynamics model



We model local population dynamics using the Ricker Model. At each

timestep, the abundance $N_i$ at location $i$ is drawn as



$$N_i(t+1) \sim \text{Poisson}\bigg(N_i(t) \lambda R e^{- \chi

N_i(t)}\bigg)$$



where $\chi$ represents the strength of mortality of surviving until

adulthood, $R$ is the probability that an adult reproduces ($0.9$ for

all results presented here), and where $\lambda$ is the mean number of

offspring for each individual that reproduces---yielding three total

parameters: $\theta = \{\lambda, R, \chi \}$.  We consider the

simplest variation on the model, which only includes demographic

stochasticity, however it is straightforward to extend this to other

forms of stochasticity [@Melbourne2008ExtRis].



## Dispersal Models



### Stochastic Dispersal



To simulate stochastic dispersal, the number of migrants leaving a

given location is stochasticly drawn each timestep  as  $m_{i} \sim

\text{Binomial}(N_i, m)$ for each location $i$. For every migrating

individual we randomly draw where that individual goes from the

distribution of potential destinations $\Phi^{(i)}$.



### Diffusion



To simulate diffusion dispersal, we incorporate the local Ricker Model

into a reaction-diffusion model. If the probability that an individual

disperses before reproducing is $m$, then we can define a diffusion

matrix $D$ as



$$D_{ij} = \begin{cases} \Phi_{ij}m \quad\quad\quad &\ i \neq j \\ 1-m

& i=j \end{cases}$$



where $D_{ij}$ is now the expected value an individual born in $i$

reproduces in $j$. The dispersal dynamics of the diffusion model are

described by the mapping



$$N_i(t+1) = \sum_j D_{ji} N_j(t)$$



which can be combined into the local Ricker model from above as

reaction-diffusion model by computing diffusion before each round of

local dynamics.



$$N_i(t+1) \sim \text{Poisson}\bigg( \lambda R e^{-\chi \big(\sum_j

D_{ji} N_j(t)\big)} \cdot \sum_j D_{ji} N_j(t) \bigg)$$



## Measuring Synchrony



In ecology and other fields, the crosscorrelation function, \(CC\), has long

been used as a measure of the synchrony between two time-series. Here, with a metapopulation, we consider the mean

crosscorrelation in abundances between all pairs of populations, which we call

Pairwise-Crosscorrelation ($\text{PCC}$) and compute as



$$\text{PCC}=\frac{1}{(N_p-1)^2}\sum_{i \neq j} CC(\vec{N_i},\vec{N_j})$$



where $\vec{N_i}$ is the time-series of abundances at population $i$.



# Results



We first consider how synchrony, measured by $\text{PCC}$, changes as

a function of the intrinsic dispersal probability $m$. In figure

@fig:migration_gradient, we see how $\text{PCC}$ changes in response

to $m$ at varying levels of both landscape connectivity $\alpha$ and

intrinsic growth rate $\lambda$. We see that under some combinations

of $\alpha$, $\lambda$, and $m$ both stochastic dispersal (green) and

diffusion (orange) produce similar levels of synchrony, however at

some parameter values diffusion artificially creates more synchronous

dynamics than stochastic dispersal.



![TODO Caption](./figures/migration_gradient_panels.png){#fig:migration_gradient}



At low $\lambda$, the diffusion model produces increasingly

synchronized population dynamics as migration increases; however, the

stochastic dispersal model produces effectively no synchrony

regardless of migration rate. As $\lambda$ increases, we see two

phenomena: 1) the distribution of $\text{PCC}$ for both diffusion and

stochastic model begin to move closer to one another, and 2) the shift

from non-synchronized to synchronized dynamics becomes more

"critical", meaning it rapidly jumps to near $\text{PCC}=1.0$ as $m$

increases. As we increase $\lambda$, the gap between the diffusion and

stochastic PCC distributions shrinks.  As $\alpha$, the modularity of

the habitat networks, increases, we see the difference in PCC between

diffusion and stochastic dispersal models shrink, but the amount of

variance in this estimate increases and we increase the modularity of

the habitat network ($\alpha$). In this case, the spatial

configuration of habitat patches, and how the dispersal structure of a

randomly generated habitat network changes with $\alpha$, is driving

greater variation in the amount of synchrony observed at a given set

of parameter values.  



To better understand this, we consider "mapping" this difference in

the parameter space defined  by varying levels of landscape

connectivity $\alpha$ and intrinsic growth rate $\lambda$ at

"snapshots" of various value of intrinsic dispersal rate $m$

(@fig:lattice). Dispersal rate is often treated as a property

intrinsic to a species.



![TODO Caption](./figures/connectivity_demography_lattice.png){#fig:lattice}



Why is it that we see a response to $\lambda$? Consider what we know

about the Ricker model,By comparing the synchrony created by

stochastic dispersal and diffusion models, we show there are two

distinct regimes: a regime where diffusion well approximates

stochastic dispersal, and a regime where it does not.



higher $\lambda$ without changing other parameters means the mean

population size increases. As the mean population size increases, the

size of the sampling distribution of dispersers at each timestep

increases, and we expect this distribution to converge to $\Phi$ as

the number of migrants increases toward infinity.



We conclude by emphasizing the difference in simulation time between

these models, especially as the number of spatial locations increases.

This is compounded by stochastic dispersal's runtime is sensitive to

the intrinsic migration probability $m$. At higher value of $m$, more

dispersal events occur,



![TODO Caption](./figures/runtime.png){#fig:runtime}



# Discussion



When developing models to understand and predict how landscape

structure effects ecological processes, diffusion can be a convenient

abstraction to speed up computation in some cases.



Here we show that diffusion can artificially synchronize dynamics

across space.



Spatial synchrony of population dynamics is generally of interest.

Dispersal induced synchrony can increase population stability, up

until a certain threshold where the dynamics become so highly

synchronized that they increase extinction risk [@Abbott2011DisPar].



The point goes beyond synchrony. The major point we intend to make

here is that if one is developing an ecological model that involves

organisms moving across space, it is imperative to test whether

stochastic and diffusion dispersal produce similar results. Diffusion

can often be a valuable abstraction that make computation faster.

"Understanding the scope and proprer domain of each abstraction"

[@Levins1987DiaBio] One way to view this is diffusion ignores temporal

variation in dispersal.



Another important consideration for this work is what is meant by a

"location" within our model. Although we frame this in terms of

habitat patches, what an individual point in a spatial network

represents is a convenient abstract to represent the spatial dimension

of ecological processes. We argue the dispersal potential, by using

probabilistic framework to represent dispersal, is a way to describe

landscape structure at any scale.



- Spatial graph models as tool for modeling ecological processes

  across space and as generative models.

- Emergent properties and the role of stochasticity



### Acknowledgments



# References