Abstract
Organisms constituting a species also include various internal heterogeneities. These include observable entities such as phenotypes to abstract states such as assets. They are not uniform for each cohort and are closely associated with critical demographic events such as reproduction and death. For instance, larger stags will have a higher degree of success in mating with females, and high-income earners will have higher survival rates because they can access advanced medical care. Although heterogeneity often inhibits the development of antibiotics and promotes social inequality, it also protects populations from extinction and triggers the evolution of species. Incorporating such individual differences in demographic theory can enable us to extend the theory to deal with a broad range of phenomena.
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Notes
- 1.
The emergence of resistant bacteria can render conventional antibiotics ineffective.
- 2.
This kernel plays role of an infinite-dimensional multi-state life table.
- 3.
We will deal with the mathematical structure of this transition kernel in the next section in this chapter.
- 4.
- 5.
This equation holds for any dimension d.
- 6.
It is called the Kolmogorov extension theorem Øksendal (2003)
- 7.
\(\hat{f*g}_{r}=\hat{f}_{r}\hat{g}_{r}\)
- 8.
If and only if all characteristic roots are not duplicated.
- 9.
It has the same meaning as the net reproduction rate in human demography. In this book, the term “basic reproduction number” is used for consistency with biological concepts.
- 10.
This age has the same meaning as the reproductive age in this type of semelparity.
- 11.
An autonomous system implies that the ingredients for the system depend on \(X_{a}\) only such that \(g_{j}\left( a,X_{a}\right) =g_{j}\left( X_{a}\right) \), \(\sigma _{jk}\left( a,X_{a}\right) =\sigma _{jk}\left( X_{a}\right) \), \(\mu \left( a,X_{a}\right) =\mu \left( X_{a}\right) \), and \(F\left( a,X_{a}\right) =F\left( X_{a}\right) \) in Eqs. (2.1), (2.2), and (2.9), respectively.
- 12.
Owing to the definition of mortality, recurrent paths do not exist such that \(K\left( 0,x^{\diamond } \rightarrow \tau ^{\prime } ,x^{\diamond }\right) =\lim _{x\uparrow x^{\diamond }}\mathcal {Q}_{\textrm{s}}\left( 0,x \rightarrow \tau ^{\prime } ,x^{\diamond }\right) \)
- 13.
This problem is to solve an ODE or a PDE under a specified value (or function) on their boundary.
- 14.
Rigorously, it is given by,
$$\begin{aligned} \psi _{r}\left( x\right) =\int _{0}^{\infty }dy \ v\left( y\right) \delta \left( x-y\right) . \end{aligned}$$ - 15.
It implies that the solution diverges at a finite time, such as
$$\begin{aligned} \lim _{t\uparrow 1}\frac{1}{1-t}=\infty . \end{aligned}$$
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Oizumi, R. (2022). Linear Structured Population Based on SDE. In: Population Dynamics Based on Individual Stochasticity. SpringerBriefs in Population Studies(). Springer, Singapore. https://doi.org/10.1007/978-981-19-3548-0_2
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