General Deterministic IPM

Abstract

This chapter is where the IPM really comes into its own as a flexible and parsimonious framework for populations with complex demography, meaning that individual state is not described by a single number z. Instead z can be multidimensional (age and size, size and disease state, etc.); it can use different attributes at different stages of the life cycle (e.g., age-structured and size-structured breeding adults, depth-structured seeds, and stage-structured plants); or the future can depend on the past as well as the present. We explain here how everything about the basic IPM continues to work in this more general setting, and discuss numerical methods for dealing with (or avoiding) the larger iteration matrices that can result when z is multidimensional. The main case study is a model with size and age structure based on the Soay sheep system, and we also use a size-quality model, and a model for tropical tree growth, to illustrate different aspects of how a general IPM is implemented numerically.

6.9 Appendix: the details

The goal of this section is to state the precise assumptions and main results of the theory for general deterministic IPMs, and give an indication of the proofs. This requires more mathematics background than the rest of the book, specifically some familiarity with functional analysis. Dunford and Schwartz (1958) is the source for assertions below that are stated without proof.

We assume that $\mathbf{Z}$ is a compact metric space, hence it is a complete and separable Hausdorff space, Borel measure is defined, and the spaces $L_{p}(\mathbf{Z}),1 \leq p <\infty$ are Banach spaces. We make $\mathbf{Z}^{2} = \mathbf{Z} \times \mathbf{Z}$ a metric space using the product metric

$$ \displaystyle{d\left ((z',z),(x',x)\right ) = \sqrt{d(z', x')^{2 } + d(z, x)^{2}}.} $$

The topology induced by this metric is the same as the product topology, so $\mathbf{Z} \times \mathbf{Z}$ is also a compact metric space.

The IPM is defined relative to a measure space $(\mathbf{Z},\mathcal{B},\mu )$ where $\mathcal{B}$ is the Borel σ-field on $\mathbf{Z}$ and μ is a finite measure on $(\mathbf{Z},\mathcal{B})$. The state of the population at time t is given by a nonnegative function n(z, t) with the interpretation that $\int \limits _{S}n(z,t)d\mu (z)$ is the number of individuals whose state at time t is in the set S. The population dynamics are defined by piecewise continuous functions K(z′, z) = F(z′, z) + P(z′, z) on $\mathbf{Z} \times \mathbf{Z}$,

$$ \displaystyle{ n(z',t + 1) =\int _{\mathbf{Z}}K(z',z)n(z,t)d\mu (z). } $$

(6.9.1)

For simplicity we use K to denote both the kernel and the linear operator (6.9.1). Piecewise continuous means that

• There is a partition of $\mathbf{Z}^{2}$ into a finite number of disjoint open sets $\mathcal{U}_{k}$ whose boundaries are a finite set of continuous curves, such that $\bigcup _{k}\bar{\mathcal{U}}_{k} = \mathbf{Z}^{2}$ where $\bar{\mathcal{U}}_{k}$ is the closure of $\mathcal{U}_{k}$.

• F and P are continuous on each $\mathcal{U}_{k}$ and can be extended as continuous functions on $\bar{\mathcal{U}}_{k}$.

When we say that F, P, or K has some pointwise property (e.g., F is positive) we mean that the property is true pointwise in all of the continuous extensions. When we say that a property holds at some point, this includes the possibility that it holds on a point of a boundary curve in one of the continuous extensions.

Properties of K depend on the domain of functions n on which it acts. It is most natural to think of K as an operator on $L_{1}(\mathbf{Z})$ (state distributions with finite total population) but we can also think of it as operating on $L_{2}(\mathbf{Z})$ and on $C(\mathbf{Z})$. Because $\mathbf{Z}$ has finite measure, $L_{2}(\mathbf{Z}) \subset L_{1}(\mathbf{Z})$ (proof: | n | ≤ 1 + | n | ², so if n ∈ L ₂ then | n | has finite integral). And because $\mathbf{Z}$ is compact any function in $C(\mathbf{Z})$ is bounded and therefore in L ₁ and L ₂. Thus $C \subset L_{2} \subset L_{1}$. Moreover, K maps L ₁ into C (proof: let n ∈ L ₁ and z _n  → z′. K is almost everywhere continuous and bounded, so the functions $f_{n}(z) = K(z_{n},z)n(z)$ are bounded above by the L ₁ function $\sup (K)n(z)$ and converge almost everywhere to f(z) = K(z′, z)n(z). The dominated convergence theorem therefore implies that Kn(z _n ) → Kn(z′)).

In short, under our assumptions K maps $L_{1} \cup L_{2} \cup C$ into $L_{1} \cap L_{2} \cap C$. As a result, an eigenvector of K in any of these spaces is in C (because an eigenvector w is a multiple of Kw which is in C). This has three important implications. First, the eigenvectors of K really are well-defined pointwise, as we have assumed in our sensitivity analysis formulas. Second, eigenvectors are in L ₁ and therefore represent finite populations. Third, the eigenvalues and eigenvectors of K are the same on all three spaces, so we can talk about eigenvalues without specifying the space.

The next issue is compactness. An operator is called compact if it maps the closed unit sphere {n: | | n | | ≤ 1} into a set with compact closure. Compactness is important because the dominant eigenvalue of a compact operator is always isolated (and the spectrum consists only of the eigenvalues and possibly 0). The dominant eigenvalue λ of any u-bounded positive operator is strictly larger in absolute value than any other eigenvalue. But that’s not enough, because there could still be an infinite sequence of eigenvalues whose absolute values converge to λ, in which case there would not be exponentially fast convergence to the stationary distribution. If that were true, the eigenvalues would have a point of accumulation on the sphere | ζ | = λ. However, only 0 can be an accumulation point for the eigenvalues of a compact operator. Thus, if K is compact, there exists 0 ≤ λ ₂ < λ such that all sub-dominant eigenvalues have absolute value ≤ λ ₂.

Because the kernel K is bounded and therefore square-integrable on $\mathbf{Z} \times \mathbf{Z}$, the operator K is compact on L ₂ (i.e., as an operator mapping L ₂ into itself; Jörgens (1982), Theorem 11.6). In Section 6.9.1 we show that K is compact on L ₁. If K is continuous rather than just piecewise continuous, then K is compact on C (this is also proved in Section 6.9.1). It is an open question (to us) whether K is necessarily compact on C if it is merely piecewise continuous.

We have stated two different positivity assumptions about K: power positivity and u-boundedness. Power-positivity is the stronger assumption because if K ^m is positive, it is u-bounded for u(z) ≡ 1. So under either assumption, some kernel iterate K ^m is compact and u-bounded. K ^m therefore satisfies the assumptions of Theorem 11.5 in Krasnosel’skij et al. (1989) in the spaces $L_{1}(\mathbf{Z})$ and $L_{2}(\mathbf{Z})$, implying that

• (a) K ^m has an eigenvalue equal to its spectral radius λ ^m, where λ is the spectral radius of K.

• (b) λ ^m is a simple eigenvalue, and the corresponding eigenvector w is the unique (up to normalization) positive eigenvector of K ^m (positive here means that w is nonnegative and nonzero).

• (c) All other eigenvalues of K ^m are have absolute value ≤ q λ ^m, for some q < 1.

Easterling (1998) showed that the three properties above also hold for K itself, with λ as the dominant eigenvalue. Stable population growth, exponentially fast convergence to stable distribution w, and the rest of stable population theory therefore follow directly from the Riesz-Schauder theory for compact operators (e.g., Zabreyko et al. (1975, p. 117) or Jörgens (1982, Section 5.7)). See Ellner and Easterling (2006) for the details, which are very similar to the finite-dimensional case.

We also want to know about v, the dominant “left eigenvector” (i.e., eigenvector of the adjoint operator K ^∗). K ^∗ has the same spectrum as K (Dunford and Schwartz 1958, VII.3.7) so λ is also the strictly dominant eigenvalue of K ^∗ and equal to its spectral radius. Because K is compact, the eigenspace decomposition for K ^∗ has the same structure same as that for K (Jörgens 1982, Section 5.7), so there is simple eigenvector v of K ^∗ corresponding to λ. K ^∗ is compact and the cone of nonnegative functions is reproducing in L ₁; under these conditions, Theorem 9.2 of Krasnosel’skij et al. (1989) says that there is an eigenvector corresponding to the spectral radius which lies in the cone. Since v is the unique eigenvector corresponding to the spectral radius λ, this tells us that v ≥ 0. Hence v is strictly positive whenever K and therefore K ^∗ are power-positive. If x is any other eigenvector (actually, any vector orthogonal to v) then

$$ \displaystyle{0 =\mathop{ \lim }\limits _{n\rightarrow \infty }\langle \lambda ^{-n}(K^{{\ast}})^{n}x,w\rangle =\langle x,\lambda ^{-n}Kw\rangle =\langle x,w\rangle } $$

so v is (up to constant multiples) the unique eigenvector of K ^∗ with $\langle v,w\rangle> 0$ (we can’t also have $\langle v,w\rangle = 0$, because in that case w would be orthogonal to everything and therefore 0).

The results above can also be applied to the next-generation operator R = F(I − P)⁻¹, to ensure existence of R ₀ as the dominant eigenvalue of R. However, the intuitive result that R ₀ − 1 and λ − 1 have the same sign has not been proved, to our knowledge, without additional assumptions. It is true in our general IPM when K and Q = (I − P)⁻¹ F are both power-positive (Ellner and Rees (2006, Appendix B), and see below). Smith and Thieme (2013, Theorem 2.4) showed that it is true when there is exact mixing at birth, F(z′, z) = c(z′)b(z). It seems that it ought to be true whenever K and R are both compact and u-bounded, but so far this has not been proved.

6.1.1 6.9.1 Derivations

1. Derivation of (6.6.1). We work here in $L_{1}(\mathbf{Z})$ using L ₁ norm for vectors and the L ₁ operator norm for operators. We assume that R satisfies our assumptions for stable population growth, hence has a positive dominant eigenvalue that equals its spectral radius ρ(R). Let $R_{m} = F(I + P + P^{2} + \cdots P^{m})$ and $E_{m} = R - R_{m} \geq 0$. We use Gelfand’s formula $\rho (A) =\mathop{ \lim }\limits _{k\rightarrow \infty }\vert \vert A^{k}\vert \vert ^{1/k}$ to bound ρ(R _m ). First, because 0 ≤ R _m  ≤ R, $\vert \vert R_{m}^{k}\vert \vert \leq \vert \vert R^{k}\vert \vert$ for all k and therefore $\rho (R_{m}) \leq \rho (R) = R_{0}$. Gelfand’s formula implies that if there exist λ ≥ 0, x ≥ 0 such that Ax ≥ λ x then ρ(A) ≥ λ. We use this to get a lower bound on ρ(R _m ). Let x ≥ 0 be the dominant right eigenvector of R, so

$$ \displaystyle{ Rx = R_{0}x,\quad R_{m}x = R_{0}x - E_{m}x. } $$

(6.9.2)

From the first equality in (6.9.2) we have Fx ≤ F(I − P)⁻¹ x = R ₀ x, therefore

$$ \displaystyle{E_{m}x = F(P^{m+1}x + P^{m+2}x + \cdots \,) \leq F(\varepsilon _{ m}x) \leq \varepsilon _{m}R_{0}x.} $$

So from the second equality in (6.9.2) $R_{m}x \geq (1 -\varepsilon _{m})R_{0}x$ and therefore $\rho (R_{m}) \geq (1 -\varepsilon _{m})R_{0}$. Hence ρ(R _m ) differs from R ₀ by at most $\varepsilon _{m}R_{0}$. □ 

2. Proof that K is compact as an operator from $L_{1}(\mathbf{Z})$ to $L_{1}(\mathbf{Z})$ . Theorem 7.1 in Luxemburg and Zaanen (1963) gives a condition for compactness of integral operators from one Banach function space to another, including operators from $L_{p}(\mathbf{Z})$ to itself under our assumptions: T is compact if and only if the set {Tn: | | n | | ≤ 1} is of uniformly absolutely continuous norm. Here | | • | | is the norm on the space where T acts, and a set of functions S is of uniformly absolutely continuous norm if for any sequence of measurable sets E _m ↓∅ and any ɛ > 0, there is an number N > 0 such that $\vert \vert f\chi _{E_{m}}\vert \vert <\varepsilon$ for all m > N and for all functions f ∈ S where χ _A is the indicator function of A (χ _A (z) = 1 for z ∈ A, 0 for z ∉ A).

Choose n ∈ L ₁(Z) with | | n | | ₁ ≤ 1. Then

$$ \displaystyle{0 \leq Kn(z') =\int _{\mathbf{Z}}K(z',z)n(z)d\mu (z) \leq sup(K)\vert \vert n\vert \vert _{1} \leq sup(K).} $$

Let ε > 0 be given, and E _m ↓∅ a sequence of measurable sets in $\mathbf{Z}$. Then

$$ \displaystyle{ \vert \vert Kn\chi _{E_{m}}\vert \vert _{1} =\int \limits _{E_{m}}\vert Kn(z')\vert d\mu (z') \leq sup(K)\mu (E_{m}). } $$

(6.9.3)

Since E _m ↓∅, we have$\mathop{\lim }\limits _{m\rightarrow \infty }\mu (E_{m}) =\mu \left (\bigcap \limits _{i=1}^{\infty }E_{i}\right ) = 0$. So given ε > 0, choose N such that $\mu (E_{m}) <\varepsilon /sup(K)$ for m > N. □ 

3. Proof that if the kernel K is continuous, K is compact as an operator on $C(\mathbf{Z})$. Because $\mathbf{Z}$ is compact any continuous K is uniformly continuous. Hence for any ε > 0 there is a δ > 0 such that $\vert K(z_{1},z) - K(z_{2},z)\vert <\epsilon$ whenever $d(z_{1},z_{2}) <\delta$. Let n be any function in the unit sphere of C, i.e., a continuous function with | n | ≤ 1 everywhere. Then if $d(z_{1},z_{2}) <\delta$,

$$ \displaystyle{\vert Kn(z_{1}) - Kn(z_{2})\vert \leq \int _{\mathbf{Z}}\vert K(z_{1},z) - K(z_{2},z)\vert \,\vert n(z)\vert \,d\mu (z) <\epsilon \mu (\mathbf{Z}).} $$

Clearly $\vert Kn\vert \leq \sup (K)\mu (\mathbf{Z})$. The image of the unit sphere is therefore equicontinuous and uniformly bounded, so by the Arzela-Ascoli Theorem it has compact closure in C, hence K is compact. □ 

4. Proof that (6.5.4) implies mixing at birth for R. If F satisfies (6.5.4), then because Rn = F(I − P)⁻¹ n, we have

$$ \displaystyle{ A\left ((I - P)^{-1}n\right )c(z') \leq Rn \leq B\left ((I - P)^{-1}n\right )c(z') } $$

(6.9.4)

as desired. And if Rn ≠ 0 it must be that (I − P)⁻¹ n ≠ 0, so A(I − P)⁻¹ n > 0 whenever Rn ≠ 0. □ 

5. Proof that R ₀ − 1 and λ − 1 have the same sign in our general IPM when Q = (I − P) ⁻¹ F and K are power-positive. Q and R have the same eigenvalues (this is equivalent to the familiar result for matrices that A and BAB ⁻¹ have the same eigenvalues for any invertible matrix B). If λ = 1 then (P + F)w = w, so Fw = (I − P)w and therefore Qw = w, so 1 is an eigenvalue of Q. Since w is a positive eigenvector of Q it must be the dominant eigenvector, hence 1 is the dominant eigenvalue of Q and therefore of R, i.e., R ₀ = 1. Reversing this argument shows that λ = 1 whenever R ₀ = 1.

Suppose that λ > 1. Then (P + F)w = λ w and therefore Fw = λ w − Pw = (λ − 1)w + (I − P)w, so

$$ \displaystyle{Qw = (I - P)^{-1}Fw = (\lambda -1)(I - P)^{-1}w + w \geq (\lambda -1)w + w =\lambda w.} $$

By theorem 9.4 of Krasnosel’skij et al. (1989) this implies that some eigenvalue of Q, and hence the dominant eigenvalue has absolute value of λ or larger, hence R ₀ > 1.

Suppose that R ₀ > 1. Then consider the family of operators K _a  = P + aF, 0 < a ≤ 1. It is easy to see that K _a and the corresponding R _a are power-positive, so there exist λ(a), R ₀(a), and so on. Because Q _a  = aQ(1), we have $R_{0}(a) = aR_{0}(1)> 1$ so there is an a < 1 such that R ₀(a) = 1 and therefore λ(a) = 1. When K is power-positive, so is aK ≤ K _a hence K _a is power-positive. Choose m so that K _a ^m is strictly positive. Then

$$ \displaystyle{K^{m+1}w_{ a}-w_{a} = K^{m+1}w_{ a}-K_{a}^{m+1}w_{ a} \geq K_{a}^{m}(Kw_{ a}-K_{a}w_{a}) = (1-a)K_{a}^{m}Fw_{ a}} $$

Since R ₀ > 0 we cannot have F = 0, and w _a is strictly positive so Fw _a is nonzero, hence $(1 - a)K_{a}^{m}Fw_{a}$ is strictly positive. It is also uniformly continuous, therefore bounded below by some positive number and therefore greater than some positive multiple of w _a . Consequently there is an ε > 0 such that $K^{m+1}w_{a} \geq (1+\epsilon )^{m+1}w_{a}$, hence the spectral radius of K ^m+1 is greater than (1 +ε)^m+1. Therefore the spectral radius of K, which is λ, must be greater than 1 +ε.

We have shown that R ₀ = 1 ⇔ λ = 1 and R ₀ > 1 ⇔ λ > 1 so we must also have R ₀ < 1 ⇔ λ < 1. □