Integrodifference equation information

In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form:

${\displaystyle n_{t+1}(x)=\int _{\Omega }k(x,y)\,f(n_{t}(y))\,dy,}$

where ${\displaystyle \{n_{t}\}\,}$ is a sequence in the function space and ${\displaystyle \Omega \,}$ is the domain of those functions. In most applications, for any ${\displaystyle y\in \Omega \,}$, ${\displaystyle k(x,y)\,}$ is a probability density function on ${\displaystyle \Omega \,}$. Note that in the definition above, ${\displaystyle n_{t}}$ can be vector valued, in which case each element of ${\displaystyle \{n_{t}\}}$ has a scalar valued integrodifference equation associated with it. Integrodifference equations are widely used in mathematical biology, especially theoretical ecology, to model the dispersal and growth of populations.^[1] In this case, ${\displaystyle n_{t}(x)}$ is the population size or density at location ${\displaystyle x}$ at time ${\displaystyle t}$, ${\displaystyle f(n_{t}(x))}$ describes the local population growth at location ${\displaystyle x}$ and ${\displaystyle k(x,y)}$, is the probability of moving from point ${\displaystyle y}$ to point ${\displaystyle x}$, often referred to as the dispersal kernel. Integrodifference equations are most commonly used to describe univoltine populations, including, but not limited to, many arthropod, and annual plant species. However, multivoltine populations can also be modeled with integrodifference equations,^[2] as long as the organism has non-overlapping generations. In this case, ${\displaystyle t}$ is not measured in years, but rather the time increment between broods.