Resolvent positive linear operators exhibit the reduction phenomenon

Lee Altenberg1

Associate Editor, BioSystems, 2605 Lioholo Place, Kihei, HI 96753-7118

Edited by Joel E. Cohen, The Rockefeller University, New York, NY, and approved December 6, 2011 (received for review August 23, 2011)

The spectral bound, sðαA þ βVÞ, of a combination of a resolvent xðt þ 1Þ¼½ð1 − αÞI þ αPDxðtÞ; [1] positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in β ∈ R. Kato's result is shown where xðtÞ is a vector of the rare allele’s frequency at time t among here to imply, through an elementary “dual convexity” lemma, that different population subdivisions, α is the rate of dispersal be- sðαA þ βVÞ is also convex in α > 0, and notably, ∂sðαA þ βVÞ∕ tween subdivisions, P is the stochastic representing the ∂α ≤ sðAÞ. Diffusions typically have sðAÞ ≤ 0, so that for diffusions pattern of dispersal, and D is a diagonal matrix of the growth rates with spatially heterogeneous growth or decay rates, greater mix- of the allele in each subdivision. The allele is protected from ing reduces growth. Models of the evolution of dispersal in parti- extinction if its asymptotic growth rate when rare is greater than cular have found this result when A is a Laplacian or second-order 1. This asymptotic growth rate is the , elliptic operator, or a nonlocal diffusion operator, implying selec- tion for reduced dispersal. These cases are shown here to be part rðAÞ≔ supfjλj: λ ∈ σðAÞg; [2] of a single, broadly general, “reduction” phenomenon. where σðAÞ is the set of eigenvalues of matrix A. perturbation theory ∣ positive semigroup ∣ reduction principle ∣ Karlin discovered that for MðαÞ≔½ð1 − αÞI þ αP, the spectral non-self-adjoint ∣ Schrödinger operator radius, rðMðαÞDÞ, is a decreasing function of the dispersal rate α, for arbitrary strongly connected dispersal pattern: he main result to be shown here is that the growth bound, TωðαA þ VÞ, of a positive semigroup generated by αA þ V Theorem 1. (Karlin's Theorem 5.2) [(16), pp. 194–196] Let P be changes with positive scalar α at a rate less than or equal to ωðAÞ, an arbitrary nonnegative irreducible stochastic matrix. Consider where A is also a generator, and V is an operator of multiplica- the family of matrices tion. Movement of a reactant in a heterogeneous environment is M α 1 − α I αP; 0 < α < 1. often of this form, where V represents the local growth or decay ð Þ¼ð Þ þ α rate, and represents the rate of mixing. Lossless mixing means Then for any diagonal matrix D with positive terms on the diagonal, ω 0 ω 0 ðAÞ¼ , while lossy mixing means ðAÞ < , so this result im- the spectral radius plies that greater mixing reduces the reactant’s asymptotic growth rate, or increases its asymptotic decay rate. Decreased growth or rðαÞ¼rðMðαÞDÞ increased decay are familiar results when A is a diffusion opera- α D ≠ I tor, so what is new here is the generality shown for this phenom- is decreasing as increases (strictly provided d ). ’ enon. At the root of this result is a theorem by Kingman on the Karlin s Theorem 5.2 means that greater mixing between sub- M α D “superconvexity” of the spectral radius of nonnegative matrices divisions produces lower rð ð Þ Þ, and if it goes below 1, the (1). The logical route progresses from Kingman through Cohen allele will go extinct. While this theorem was motivated by the (2) to Kato (3). The historical route begins in population genetics. issue of genetic diversity in a subdivided population, its form ap- In early theoretical work to understand the evolution of genet- plies generally to situations where differential growth is combined D ic systems, Feldman, colleagues, and others kept finding a com- with mixing. could just as well represent the investment returns P D mon result from each model they examined (4–14)—be they on different assets and a pattern of portfolio rebalancing. Or models for the evolution of recombination, or of mutation, or could represent the decay rates of reactant in different parts of a P of dispersal. Evolution favored reduced levels of these processes reactor, and a pattern of stirring within the reactor. In a very in populations near equilibrium under constant environments, general interpretation, Theorem 5.2 means that greater mixing and this result was called the Reduction Principle (11). reduces growth and hastens decay.

α These results were found for finite-dimensional models. But If the dispersal rate is not an extrinsic parameter, but is a the same reduction result has also been found in models for variable which is itself controlled by a gene, then a gene which α the evolution of unconditional dispersal in continuous space, in decreases will have a growth advantage over its competitor which matrices are replaced by linear operators. This finding alleles. The action of such modifier genes produces a process raises the questions of whether this common result, discovered that will reduce the rates of dispersal in a population. Therefore, in such a diversity of models, reflects a single mathematical phe- Theorem 5.2 also means that differential growth selects for reduced nomenon. Here, the question is answered affirmatively. mixing. The mathematical underpinnings of the reduction principle In the evolutionary context, the generality of the mixing pat- P ’ for finite-dimensional models were discovered by Sam Karlin tern in Karlin s Theorem 5.2 makes it applicable to other kinds POPULATION BIOLOGY (15, 16) [although he did not realize it, and he had earlier pro- posed an alternate to the reduction principle—the mean fitness Dedicated to Sir John F. C. Kingman on the fiftieth anniversary of his theorem on the principle (17), which was found to have counterexamples (18)]. “superconvexity” of the spectral radius (1), which is at the root of the results presented Karlin wanted to understand the effect of population subdivision here. on the maintenance of genetic variation. Genetic variation is Author contributions: L.A. designed research, performed research, and wrote the paper. preserved if an allele has a positive growth rate when it is rare, The author declares no conflict of interest. protecting it from extinction. The dynamics of a rare allele are This article is a PNAS Direct Submission. approximately linear, and of the form 1E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1113833109 PNAS ∣ March 6, 2012 ∣ vol. 109 ∣ no. 10 ∣ 3705–3710 Downloaded by guest on September 27, 2021 of “mixing” besides dispersal. The pattern matrix P can just as BðXÞ represents the set of all bounded linear operators well refer to the pattern of mutations between genotypes, and A: X → X. α P ⊂ then refers to the mutation rate. Or can represent the pattern A is a positive operator if AXþ X þ. of transmission when two loci recombine, and then α represents The resolvent of A is Rðξ;AÞ≔ðξ − AÞ−1, the operator inverse of the recombination rate. The early models for the evolution of ξ − A, ξ ∈ C. recombination and mutation that exhibited the reduction princi- The resolvent set ρðAÞ ⊂ C are those values of ξ for which ple in fact had the same form as Eq. 1 for the dynamics of a rare ξ − A is invertible. modifier allele. Once this commonality of form was recognized The spectrum of A ∈ BðXÞ, σðAÞ, is the complement of the (19–21), it was clear that Karlin’s theorem explained the repeated resolvent set, ρðAÞ. appearance of the reduction result in the different contexts, and The spectral bound of closed linear operator A, not necessarily generalized the result to a whole class of genetic transmission bounded, is patterns beyond the special cases that had been analyzed. The dynamics of movement in space have been long modeled supfReðλÞ: λ ∈ σðAÞg if σðAÞ ≠ ∅ sðAÞ≔ : by infinite-dimensional models, where space is continuous and −∞ if σðAÞ¼∅ the concentrations of a quantity at each point are represented as a function. The dynamics of change in the concentration are The type (growth bound) of an infinitesimal generator, A,ofa modeled as diffusions, where the Laplacian or elliptic differential tA 0 operator or nonlocal integral operator takes the place of the strongly continuous (C0) semigroup, fe : t> g,is matrix P in the finite-dimensional case. When the substance 1 grows or decays at rates that are a function of its location, the ωðAÞ≔lim log ∥etA∥ ¼ log rðeAÞ: t→∞ t system is often referred to as a reaction diffusion. In reaction- diffusion models for the evolution of dispersal, the reduction Generally, −∞ ≤ sðAÞ ≤ ωðAÞ < ∞, but conditions for principle again makes its appearance (22) (23, Lemma 5.2) (24, sðAÞ¼ωðAÞ or sðAÞ < ωðAÞ are part of a more involved theory Lemma 2.1) (25). In nonlocal diffusion models, again the reduc- for the asymptotic growth of semigroups (see refs. 36–38). tion principle appears (26). This repeated occurrence points to the possibility of an underlying mathematical unity. Here, a broad characterization of this “reduction phenomen- Definition 1: Operator A is resolvent positive if there is ξ0 such that on” is established by generalizing Karlin’s theorem to linear ðξ0;∞Þ ⊂ ρðAÞ and Rðξ;AÞ is positive for all ξ > ξ0 (39). operators. The reduction results previously found for various The relationship of the resolvent positive property to other linear operators are, therefore, seen to be special cases of a familiar operator properties includes the following list of key general phenomenon. This result is actually implicit in Kato’s generalization (3) of results: Cohen’s theorem (2) on the convexity of the spectral bound of ≥ 1. If A generates a C0-semigroup Tt, then Tt is positive for all t essentially nonnegative matrices with respect to the diagonal 0 if and only if A is resolvent positive (ref. 38, p. 188). elements of the matrix. It is educed from Kato’s theorem here 2. If A is a resolvent positive operator defined densely on by means of an elementary “dual convexity” lemma. X ¼ CðSÞ, the of continuous complex-valued Kato’s goal in ref. 3 was to generalize, from matrices to linear functions on compact space S, then A generates a positive operators, Cohen’s convexity result (2): C0-semigroup [(38), Theorem 3.11.9]. 3. If A is resolvent positive and its domain, DðAÞ ⊂ X, is dense in 2 Theorem 2. (Cohen) (2) Let D be diagonal real n × n matrix. Let A be X, then for every f ∈ DðA Þ, there exists a unique solution, 1 an essentially nonnegative n × n matrix. Then sðA þ DÞ is a convex uðtÞ ∈ DðAÞ for all t ≥ 0, u ∈ C ð½0;∞Þ;XÞ, to the Cauchy function of D. problem (39, Theorem 7.1) ∂u A D — ¼ AuðtÞðt ≥ 0Þ;uð0Þ¼f: Here, sð þ Þ is the spectral bound the largest real part ∂t of any eigenvalue of A þ D. A synonym for the spectral bound used in the matrix literature is the spectral abscissa (27, 28). When 4. If A is resolvent positive then: sðAÞ < ∞;ifσðAÞ is nonempty; the spectral bound is an eigenvalue, it is also referred to as the i.e., −∞ < sðAÞ, then sðAÞ ∈ σðAÞ;ifξ ∈ R ∩ ρðAÞ yields principal eigenvalue (29), dominant eigenvalue (30), dominant root Rðξ;AÞ ≥ 0 then ξ >sðAÞ (3) (38, Proposition 3.11.2). (31), Perron-Frobenius eigenvalue (32), or Perron root (33). “Essen- 5. Differential operators higher than second order are never tially nonnegative” means that the off-diagonal elements are non- resolvent positive (40, Corollary 2.3) (41). negative. Synonyms include “quasi-positive” (34), “Metzler,” 6. Well-known examples of resolvent positive operators include “Metzler-Leontief,”“ML” (32), and “cooperative” (35). the following (for details see the sample references): Cohen’s proof relied upon the following theorem of Kingman: 1 p N Schrödinger operators − 2 Δ þ V on L ðR Þ, where Δ ¼ ∑N ∂2∕∂x2 is the Laplace operator, and V is an operator A × i¼1 i Theorem 3. (Kingman) (1) Let be an n n matrix whose elements, of multiplication with constraints depending on p (see θ θ Aijð Þ, are nonnegative functions of the real variable , such that they refs. 42–44). “ ” θ θ are superconvex, i.e., for each i, j, either log Aijð Þ is convex in Second-order elliptic operators on LpðΩÞ, θ θ 0 θ [Aijð Þ is log convex], or Aijð Þ¼ for all . Then the spectral ra- A θ dius of is also superconvex in . N ∂ ∂ N ∂ ∂ A ¼ ∑ ∂ ajk ∂ þ ∑ bj ∂ þ ∂ ðcj:Þþa0; xk xj xj xj Kato generalized Cohen’s result to linear operators by first j;k¼1 j¼1 generalizing Kingman’s theorem. Before presenting Kato’s theo- rem, some terminology needs to be introduced: where Ω ⊂ RN is open, coefficients are measurable and X represents an ordered Banach space or its complexification. bounded, ellipticity conditions apply to ajkðxÞ, and appropriate Xþ represents the proper, closed, positive cone of X, assumed additional conditions hold for the coefficients, domain and to be generating and normal (see ref. 3). boundary (45), also e.g., (3, 46, 47).

3706 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1113833109 Altenberg Downloaded by guest on September 27, 2021 ∈ D 0 ∞ ∈ D Linear integral operators A on X ¼ CðΩ¯Þ defined by Lemma 1. (Dual Convexity) Let x 1 ¼ð ; Þ and y 2 ¼ Z ½0;∞Þ. Let f: D1 × D2 → R have the following properties: ðAfÞðxÞ≔ Kðx;yÞfðyÞdy þ bðxÞfðxÞ; fðαx;αyÞ¼α fðx;yÞ; for α > 0; and [6] Ω ∈ Ω¯ × Ω¯ Rþ Ω ⊂ RN 0 where K Cð ; Þ, is bounded, and Kðx;yÞ > , fðx;yÞ is convex in y: [7] bðxÞ are measurable functions for x, y ∈ Ω¯ (26, 48, 49). A resol- vent positive combination of integral and differential operator Then: is analyzed in ref. 50. 1. fðx;yÞ is convex in x; Kato’s generalization of Cohen’s theorem is as follows: 2. For each x ∈ D1, either 1 0 ∀ ∈ D Theorem 4. (Generalized Cohen’s Theorem) (3) Consider X C S (a) fðx þ d;yÞ < fðx;yÞþdfð ; Þ d 1;or ¼ ð Þ 1 0 ∀ ∈ D (continuous functions on a compact Hausdorff space S)or (b) fðx þ d;yÞ¼fðx;yÞþdfð ; Þ d 1. X ¼ LpðSÞ, 1 ≤ p < ∞, on a measure space S, or more generally, For y ≠ 0,iffðx;yÞ is strictly convex in y, then fðx;yÞ is strictly con- let X be the intersection of two Lp-spaces with different p’s and vex in x, and fðx þ d;yÞ < fðx;yÞþdfð1;0Þ. different weight functions. Let A: X → X be a linear operator which 3. For each x ∈ D1, is resolvent positive. Let V be an operator of multiplication on X ∂ represented by a real-valued function v, where v ∈ CðSÞ for fðx;yÞ ≤ fð1;0Þ; ∂x X ¼ CðSÞ,orv ∈ L∞ðSÞ for the other cases. Then sðA þ VÞ is a convex function of V. If in particular A is a generator of a C0 semi- except possibly at a countable number of points x, where the group, then both sðA þ VÞ and ωðA þ VÞ are convex in V. one-sided derivatives exist but differ: ∂ ∂ Kato’s theorem is further generalized to Banach lattices by fðx;yÞ < fðx;yÞ ≤ fð1;0Þ: ∂x− ∂x Arendt and Batty (51) as follows: þ

The lemma holds if we substitute D1 ¼ð−∞;0Þ or D2 ¼ Theorem 5. (Generalized Kato’s Theorem) [(51), Theorem 3.5] Let A ð−∞;0 or both. be the generator of a positive semigroup on a Banach lattice X, and let ZðXÞ≔fT ∈ LðXÞ: ∃c ≥ 0;jTxj ≤ cjxjðx ∈ XÞg refer to the “cen- ” ↦ ↦ ω ter of X.ThenthefunctionsV sðA þ VÞ and V ðA þ VÞ Proof: from ZðXÞ into ½−∞;∞Þ are convex. 1. fðx;yÞ is convex in x. The relation fðαx;αyÞ¼α fðx;yÞ (f is homogeneous of degree Results one) allows a set of rescalings that transform convexity in y into convexity in x. It is perhaps worth noting that this relation is Theorem 6. (Generalized Karlin’s Theorem) Let A be a resolvent actually a homomorphism, which can be put into a more fa- positive linear operator, and V be an operator of multiplication, miliar form by defining a product x⋆y ≔fðx;yÞ, and function ψ ≔α ψ ⋆ψ ψ ⋆ under the same assumptions as Theorem 4 (or let A and V be as ðxÞ x, which gives ðxÞ ðyÞ¼ ðx yÞ. 0 6 α 0 α 0 in Theorem 5). Then for α > 0 : For the case y ¼ , Eq. gives fð x; Þ¼ fðx; Þ,sof is trivially convex in x. 1. sðαA þ VÞ is convex in α; For y ≠ 0, the following derivations have the constraints 2. For each α > 0, either y, y1, y2 ∈ D2, y, y1, y2 ≠ 0, and 0 < m < 1, so that fy;y1;y2;m;1 − m;ð1 − mÞy1 þ my2g are nonzero and their sððα þ dÞA þ VÞ < sðαA þ VÞþdsðAÞ ∀ d>0,or α α ∀ 0 ratios and reciprocals are always defined, and ratios of yi sðð þ dÞA þ VÞ¼sð A þ VÞþdsðAÞ d> ; terms always positive. These constraints keep the arguments 3. In particular, when sðAÞ¼0 then sðαA þ VÞ is nonincreasing α 0 of f within their domains throughout the rescalings. in (the reduction phenomenon), and when sðAÞ < then Convexity of fðx;yÞ in y gives sðαA þ VÞ is strictly decreasing in α; 4. For each α > 0, ð1 − mÞfðx;y1Þþmfðx;y2Þ ≥ fðx;ð1 − mÞy1 þ my2Þ; [8] ∈ 0 1 ≠ 6 8 d α ≤ [3] for m ð ; Þ, y1 y2. Applying [ ]to[], with respective sð A þ VÞ sðAÞ; α ∕ α ∕ α 1 − ∕ dα substitutions ¼ y1 y, ¼ y2 y, and ¼½ð mÞy1 þ my2 y MATHEMATICS in the three f terms, yields: except possibly at a countable number of points α, where the one-sided derivatives exist but differ: y1 xy y2 xy ð1 − mÞ f ;y þ m f ;y d d y y1 y y2 sðαA þ VÞ < sðαA þ VÞ ≤ sðAÞ: [4] dα− dα þ ð1 − mÞy1 þ my2 xy ≥ f ;y : [9] If A is a generator of a C0-semigroup, then the above relations on y ð1 − mÞy1 þ my2 sðαA þ VÞ also apply to the type ωðαA þ VÞ.

Let x1≔xy∕y1 and x2≔xy∕y2 represent the rescaled first argu- POPULATION BIOLOGY Proof: We consider the general form ments for f on the left side of [9] (so x, x1, x2 ∈ D1). We try the ansatz that x1 and x2 can be combined convexly to ϕðα;βÞ≔sðαA þ βVÞ or ωðαA þ βVÞ; [5] yield the third rescaled argument on the right side of [9]: where α > 0, β ∈ R. Kato (3) explicitly shows that ϕð1;βÞ is convex xy xy xy β ¼ð1 − hÞx1 þ hx2 ¼ð1 − hÞ þ h : in (which he points out is equivalent to varying V). Kato's result ð1 − mÞy1 þ my2 y1 y2 is shown to imply the properties claimed for sðαA þ VÞ¼ϕðα;1Þ with respect to variation in α, by Lemma 1, to follow. The ansatz has solution

Altenberg PNAS ∣ March 6, 2012 ∣ vol. 109 ∣ no. 10 ∣ 3707 Downloaded by guest on September 27, 2021 my2 ð1 − mÞy1 d3 − d2 d2 − d1 h ¼ ; and 1 − h ¼ : gðx þ d2Þ ≤ gðx þ d1Þþ gðx þ d3Þ ð1 − mÞy1 þ my2 ð1 − mÞy1 þ my2 d3 − d1 d3 − d1 d3 − d2 d2 − d1 Note that h ∈ ð0;1Þ is assured because y1 and y2 have the same < ðgðxÞþd1FÞþ gðx þ d3Þ d3 − d1 d3 − d1 sign, y1 ≠ y2, and m ∈ ð0;1Þ. − − Define ϕ≔½ð1 − mÞy1 þ my2∕y. Then ϕ > 0 because y, y1, and d3 d2 d2 d1 ≤ ðgðxÞþd1FÞþ ðgðxÞþd3FÞ y2 all have the same sign. Substitution gives ð1 − mÞy1∕y ¼ d3 − d1 d3 − d1 ð1 − hÞϕ, and my2∕y ¼ hϕ, and [9] becomes: ¼ gðxÞþd2F:

ð1 − hÞϕfðx1;yÞþhϕfðx2;yÞ ≥ ϕfðð1 − hÞx1 þ hx2;yÞ: For the case where D1 ¼ð−∞;0Þ, the direction of inequalities 14 After dividing both sides by ϕ > 0, in [ ] needs to be reversed, and all the subsequent relations are preserved. 3. For each x ∈ D1, ∂fðx;yÞ∕∂x ≤ fð1;0Þ, except possibly at a ð1 − hÞfðx1;yÞþhfðx2;yÞ ≥ fðð1 − hÞx1 þ hx2;yÞ; [10] countable number of points x, where the one-sided derivatives ∂fðx;yÞ ∂fðx;yÞ ∃ exist but differ: ∂ < ∂ ≤ fð1;0Þ. which is convexity in x [for each ðx1;x2;hÞ; ðy1;y2;mÞ]. The case x− xþ of strict convexity follows by substituting > for ≥ throughout. Rearrangement of [11] gives 2. Either fðx þ d;yÞ < fðx;yÞþdfð1;0Þ ∀ d ∈ D1,orfðx þ d;yÞ¼ fðx;yÞþdfð1;0Þ ∀ d ∈ D1. fðx þ d;yÞ − fðx;yÞ ≤ fð1;0Þ; so [15] If y ¼ 0, then case 2b in Lemma 1 holds by [6]: d

fðx þ d;0Þ¼ðx þ dÞfð1;0Þ¼fðx;0Þþdfð1;0Þ: fðx þ d;yÞ − fðx;yÞ ∂fðx;yÞ lim ≕ ≤ fð1;0Þ: [16] ↓0 ∂ For y ≠ 0, the strategy will be to show first that d d xþ fðx þ d;yÞ ≤ fðx;yÞþdfð1;0Þ. Next, it is shown that if fðx þ d;yÞ < fðx;yÞþdfð1;0Þ for any d ∈ D1, then it is true For y ¼ 0, equality holds in [15] for all d>0.Fory ≠ 0, for all d ∈ D1. because fðx;yÞ is convex in x on the open interval D1, the The steps are shown here only for x, d ∈ D1 ¼ð0;∞Þ, but they left-sided and right-sided derivatives always exist, and differ are readily applied to D1 ¼ð−∞;0Þ.By[6], for x, d>0, the at most at a countable number of points, at which the right- following are equivalent: sided derivative [16] is greater than the left-sided derivative [(52) Proposition 17, pp. 113–114]. fðx þ d;yÞ ≤ fðx;yÞþdfð1;0Þ; [11] A concavity version of the lemma may be trivially produced by y y reversal of the convexity inequalities. ðx þ dÞf 1; ≤ xf 1; þ dfð1;0Þ; and x þ d x y x y d Remark 1: It would be clearly desirable to characterize the condi- f 1; ≤ f 1; þ fð1;0Þ: [12] tions for strict convexity in Kato’s theorem, so that by Lemma 1, x þ d x þ d x x þ d one would obtain strict convexity in Theorem 6, item 1, and strict monotonicity in items 3 and 4. Item 2 is the best that can be Because 0 < d∕ðx þ dÞ, x∕ðx þ dÞ < 1, the second arguments 12 offered in the way of strict inequality without strict convexity. But for f in [ ] are related by convex combination, the problem is more technical and is deferred to elsewhere. It is reasonable, nevertheless, to conjecture that the properties y x y x ¼ þ 1 − 0; which produce strict convexity in the matrix case (ref. 53, Theo- x þ d x þ d x x þ d rem 4.1) (ref. 54, Theorem 1.1) extend to their Banach space versions: i.e., for a>0, when resolvent positive operator A is so [12] is just a statement of the convexity of fðx;yÞ in y,as irreducible (ref. 55, p. 250) (ref. 51, p. 41), then sðαA þ βVÞ is hypothesized. Strict convexity of fðx;yÞ in y replaces ≤ with strictly convex in β if and only if V is not a constant scalar. < throughout [11] and [12], yielding case 2a in Lemma 1. The conjectured sharpening of Theorem 6 to strict inequality Now, with x>0, suppose that for some d1 > 0, would have application to continuous-space models for the evo- lution of dispersal, and show populations to be invadable by fðx þ d1;yÞ < fðx;yÞþd1fð1;0Þ: [13] less-dispersing organisms when they experience spatially hetero- geneous growth rates (a “selection potential” as defined in We shall see that convexity then prevents fðx þ d;yÞ from ever ref. 20). This invasibility result is a key element of the Reduction returning to the line fðx;yÞþdfð1;0Þ for d>0. Principle, and is first stated generally for finite matrix models in We consider five points: refs. 19, pp. 118, 126, 137, 195, 199 and 20, Results 2, 3. The invasibility result's primary implication is that for a population 0 < x < x þ d0 < x þ d1 < x þ d2 < x þ d3: [14] to be non-invadable, it must experience no spatial heterogeneity of growth rates where it has positive measure, and this points to- For readability, write gðxÞ ≡ fðx;yÞ and F ≡ fð1;0Þ. By convexity ward ideal free distributions (defined to be those which spatially [10], and hypothesis [13], equalize the growth rates when this is possible), as the evolutio- narily stable states. For reviews and recent developments, see d0 d0 refs. 56–60. gðx þ d0Þ ≤ 1 − gðxÞþ gðx þ d1Þ d1 d1 A Third Proof of Karlin’s Theorem 5.2. Karlin’s proof is based on the d0 d0 Donsker-Varadhan variational formula for the spectral radius < 1 − gðxÞþ ðgðxÞþd1FÞ¼gðxÞþd0F; d1 d1 (62). Kirkland, et al. (56) recently discovered another proof using entirely structural methods. A third distinct proof of Karlin’s and, by [10], [13], and [11] (line 3 below), theorem is seen as follows by application of Lemma 1 to Cohen’s

3708 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1113833109 Altenberg Downloaded by guest on September 27, 2021 theorem, combined with Friedland’s equality condition [(53), diagonal matrix. Then Theorem 4.1] (see also refs. 53 and 62 for other proof methods). The expression in Karlin’s Theorem 5.2 can be put into the sðA þ DÞ − sðAÞ ≥ uðAÞ⊤DvðAÞ: [18] form used in Theorem 6: Proof: Because A is an irreducible essentially nonnegative matrix, MðαÞD ¼½ð1 − αÞI þ αPD ¼ αðP − IÞD þ D ¼ αA þ βD; sðAÞ is an eigenvalue of multiplicity 1. Consider the representa- tion A ¼ αB − D, where B is essentially nonnegative and α > 0. where A ¼ðP − IÞD, α ∈ ð0;1Þ, and β ¼ 1. Because MðαÞD is a α ∈ 0 1 Write s ≡ sðAÞ. Because A is irreducible, it has unique u ≡ nonnegative matrix when ð ; Þ, by Perron-Frobenius theory u A v ≡ v A u⊤v e⊤v 1 its spectral bound sðMðαÞDÞ equals its spectral radius rðMðαÞDÞ. ð Þ and ð Þ given ¼ ¼ , and all the derivatives Cohen’s theorem gives that sðαA þ βDÞ is convex in β, and thus exist (28) in the following derivation [(66), Sec. 9.1.1]: by Lemma 1, sðαA þ βDÞ is convex in α > 0 and ∂ðAvÞ ∂A ∂v ∂v ∂ u⊤ ¼ u⊤ v þ A ¼ u⊤Bv þ su⊤ ¼ u⊤ ðsvÞ ∂sðαA þ βDÞ ∂α ∂α ∂α ∂α ∂α ≤ sðAÞ¼sððP − IÞDÞ¼0; [17] ∂α ∂ ∂v ∂ ∂v u⊤ s v s u⊤ ¼ ∂α þ s ∂α ¼ ∂α þ s ∂α : the right identity seen because e⊤ðP − IÞD ¼ðe⊤ − e⊤ÞD ¼ 0, e e⊤ where is the vector of ones, and is its transpose. u⊤∂v∕∂α Strict convexity in β is shown by Friedland (ref. 53, Theorem Cancellation of terms s gives 4.1) to occur when P is irreducible and D ≠ cI, for any c>0. ∂sðAÞ ∂A Strict convexity in β implies, by Lemma 1, that rðMðαÞDÞ¼ ¼ uðAÞ⊤ vðAÞ¼uðAÞ⊤BvðAÞ ≤ sðBÞ; sðMðαÞDÞ is strictly convex and decreasing in α. ∂α ∂α the inequality derived as in Eq. 17. Scaling by α, subtracting D, and substituting αB ¼ A þ D, we get Remark 2: The core of Kirkland, et al.’s (56) proof of Theorem 1 is their Lemma 4.1, which can be expressed as uðAÞ⊤ðαB − DÞvðAÞ¼sðAÞ ≤ sðαBÞ − uðAÞ⊤DvðAÞ ⊤ ⊤ e AðuðAÞ ∘ vðAÞÞ ≥ uðAÞ AvðAÞ¼sðAÞ; ⇔ uðAÞ⊤DvðAÞ ≤ sðA þ DÞ − sðAÞ:

with equality only when e⊤A ¼ sðAÞe⊤, where uðAÞ⊤ and vðAÞ are A Key Open Problem. In some physical systems, and in biological the left and right eigenvectors of A associated with the Perron applications especially, there may be multiple, independently var- root sðAÞ, and u ∘ v is the Schur-Hadamard (elementwise) ied operators acting on a quantity [e.g., diffusion with indepen- product. dent advection (ref. 67, eq. 2.9)], or the variation may not scale Kirkland, et al.’s (55) result, except for the equality condition, the mixing process uniformly [e.g., conditional dispersal (68, 69)], is a special case of ref. 63 Theorem 3.2.5 that x⊤Ay ≥ sðAÞ for any so that variation is not of the form αA þ V but rather αA þ B, x;y ≥ 0:x ∘ y ¼ uðAÞ ∘ vðAÞ. To obtain the equality condition re- where B is a linear operator other than an operator of multiplica- quires an approach Kirkland, et al.’s (55) distinct proof provides. tion. Examples are known where departures from reduction oc- cur; i.e., dsðαA þ BÞ∕dα >sðAÞ. Results for the form αA þ B have Remark 3: Schreiber and Lloyd-Smith [(64) Appendix B, Lemma been obtained for symmetrizable finite matrices in models of 1] followed the reverse path and extended Kirkland, et al’s result multilocus mutation (70), and dispersal in random environments on sðMðαÞDÞ to the form sðαA þ DÞ, where A is essentially non- (71). Some results for Banach space models have also been ob- negative and D any diagonal matrix. tained (67–69, 72–77). A key open problem, then, is to find necessary or sufficient ∂ α β ∕ Remark 4: Kato (3) notes that the Donsker-Varadhan formula pro- conditions on Banach space operators, B, such that sð A þ BÞ ∂α ≤ β α vides another route besides Kingman’s theorem to his generali- sðAÞ (which may depend on A, / , domain, and boundary α β zation of Cohen’s theorem (but with more restrictive conditions). conditions). A sufficient condition is that sð A þ BÞ be convex β Indeed, Friedland (53) uses the Donsker-Varadhan formula to in , by Lemma 1. Thus, the dual problem is to ask: for which prove Cohen’s theorem augmented by strict convexity. The dual B is sðαA þ βBÞ convex in β? Some results towards this problem ’ ’ are in ref. 78. Kato obtained Theorem 4 with operators of convexity relationship shown here between Cohen s and Karlin s β theorems means that both routes of proof apply as well to Karlin’s multiplication, V, because the family of operators e V is semi- β theorem. Given these parallels, the relationship between the group-superconvex in (definition in ref. 3), but this semi- MATHEMATICS theorem of Kingman and the theorem of Donsker and Varadhan group-superconvexity approach faces the challenge that, “It is invites deeper study. in general difficult to find a nontrivial semigroup-superconvex Lemma 1 combined with Cohen’s theorem can also be used to family B(h)” (3). give a new proof of an inequality of Lindqvist, the special case considered in ref. 65, Theorem 2, pp. 260–261. ACKNOWLEDGMENTS. I thank Prof. Shmuel Friedland for introducing me to the papers of Cohen, and for inviting me to speak on the early state of this Theorem 7. (Lindqvist) [(65), Theorem 2, subcase ] Let A be an work at the 16th International Society Meeting in Pisa; × ≥ 0 ≠ Prof. Mustapha Mokhtar-Kharroubi and anonymous reviewers for pointing irreducible n n real matrix such that (i) Aij for i j, and ⊤ out an error in a definition in the first version; anonymous reviewers for (ii) The left and right eigenvectors of A, uðAÞ and vðAÞ, associated helpful suggestions; Laura Marie Herrmann for assistance with the literature POPULATION BIOLOGY ⊤ with eigenvalue sðAÞ, satisfy uðAÞ vðAÞ¼1. Let D be an n × n real search; and Arendt and Batty (51) for guiding me to Kato (3).

1. Kingman JFC (1961) A convexity property of positive matrices. Q J Math 12:283–284. 5. Feldman MW (1972) Selection for linkage modification: I. Random mating popula- 2. Cohen JE (1981) Convexity of the dominant eigenvalue of an essentially nonnegative tions. Theor Popul Biol 3:324–346. – matrix. P Am Math Soc 81:657 658. 6. Balkau B, Feldman MW (1973) Selection for migration modification. Genetics 3. Kato T (1982) Superconvexity of the spectral radius, and convexity of the spectral 74:171–174. bound and the type. Mathematische Zeitschrift 180:265–273. 7. Karlin S, McGregor J (1974) Towards a theory of the evolution of modifier genes. Theor 4. Gadgil M (1971) Dispersal: population consequences and evolution. Ecology Popul Biol 5:59–103. 52:253–261.

Altenberg PNAS ∣ March 6, 2012 ∣ vol. 109 ∣ no. 10 ∣ 3709 Downloaded by guest on September 27, 2021 8. Feldman MW, Krakauer J (1976) Genetic modification and modifier polymorphisms. 44. Mokhtar-Kharroubi M (2009) On Schrödinger semigroups and related topics. J Funct Population Genetics and Ecology, eds S Karlin and E Nevo (Academic Press, New York), Anal 256:1998–2025. pp 547–583. 45. Ouhabaz EM (2005) Analysis of Heat Equations on Domains, London Mathematical 9. Teague R (1976) A result on the selection of recombination altering mechanisms. Society Monograph Series, (Princeton University Press, Princeton, NJ), 31. J Theor Biol 59:25–32. 46. Donsker M, Varadhan S (1976) On the principal eigenvalue of second-order elliptic 10. Teague R (1977) A model of migration modification. Theor Popul Biol 12:86–94. differential operators. Commun Pur Appl Math 29:595–621. 11. Feldman MW, Christiansen FB, Brooks LD (1980) Evolution of recombination in a 47. Berestycki H, Nirenberg L, Varadhan S (1994) The principal eigenvalue and maximum constant environment. Proc Nat'l Acad Sci USA 77:4838–4841. principle for second-order elliptic operators in general domains. Commun Pur Appl 12. Holt R (1985) Population dynamics in two-patch environments: some anomalous Math 47:47–92. consequences of an optimal habitat distribution. Theor Popul Biol 28:181–208. 48. Grinfeld M, Hines G, Hutson V, Mischaikow K, Vickers G (2005) Non-local dispersal. 13. Feldman MW, Liberman U (1986) An evolutionary reduction principle for genetic Differential and Integral Equations (Athens) 18:1299–1320. modifiers. Proc Nat'l Acad Sci USA 83:4824–4827. 49. Bates P, Zhao G (2007) Existence, uniqueness and stability of the stationary solution to 14. McPeek M, Holt R (1992) The evolution of dispersal in spatially and temporally varying a nonlocal evolution equation arising in population dispersal. J Math Anal Appl environments. Am Nat 140:1010–1027. 332:428–440. 15. Karlin S (1976) Population subdivision and selection migration interaction. Population 50. Chabi M, Latrach K (2002) On singular mono-energetic transport equations in slab Genetics and Ecology, eds S Karlin and E Nevo (Academic Press, New York), pp 616–657. geometry. Math Method Appl Sci 25:1121–1147. 16. Karlin S (1982) Classifications of selection–migration structures and conditions for a 51. Arendt W, Batty CJK (1995) Principal eigenvalues and perturbation. Operator Theory: protected polymorphism. Evolutionary Biology, eds MK Hecht, B Wallace, and Advances and Applications 75:39–55. GT Prance (Plenum Publishing Corporation, NY), 14, pp 61–204. 52. Royden HL (1988) Real Analysis (Macmillan, New York), 3rd Ed.. 17. Karlin S, McGregor J (1972) The evolutionary development of modifier genes. Proc 53. Friedland S (1981) Convex spectral functions. Linear Multilinear A 9:299–316. Nat'l Acad Sci USA 69:3611–3614. 54. Nussbaum D (1986) Convexity and log convexity for the spectral radius. Linear Algebra 18. Karlin S, Carmelli D (1975) Numerical studies on two-loci selection models with general Appl 73:59–122. viabilities. Theor Popul Biol 7:399–421. 55. Greiner G, Voigt J, Wolff M (1981) On the spectral bound of the generator of 19. Altenberg L (1984) A Generalization of Theory on the Evolution of Modifier Genes semigroups of positive operators. J Operat Theor 5:245–256. (ProQuest, Stanford University). 56. Kirkland S, Li CK, Schreiber SJ (2006) On the evolution of dispersal in patchy land- 20. Altenberg L, Feldman MW (1987) Selection, generalized transmission, and the evolu- scapes. SIAM J Appl Math 66:1366–1382. tion of modifier genes. I. The reduction principle. Genetics 117:559–572. 57. Cantrell R, Cosner C, Deangelis D, Padron V (2007) The ideal free distribution as an 21. Altenberg L (2009) The evolutionary reduction principle for linear variation in genetic evolutionarily stable strategy. Journal of Biological Dynamics 1:249–271. transmission. B Math Biol 71:1264–1284. 58. Cantrell R, Cosner C, Lou Y (2010) Evolution of dispersal and the ideal free distribution. 22. Hastings A (1983) Can spatial variation alone lead to selection for dispersal? Theor Math Biosci Eng 7:17–36. Popul Biol 24:244–251. 59. Averill I, Lou Y, Munther D (2011) On several conjectures from evolution of dispersal. 23. Hutson V, López-Gómez J, Mischaikow K, Vickers G (1995) Limit behavior for a com- Journal of Biological Dynamics. peting species problem with diffusion. Dynamical Systems and Applications,ed 60. Cosner C, Dávila J, Martínez S (2011) Evolutionary stability of ideal free nonlocal RP Agarwal (World Scientific, Singapore), pp 343–358. dispersal. Journal of Biological Dynamics. 24. Dockery J, Hutson V, Mischaikow K, Pernarowski M (1998) The evolution of slow 61. Donsker MD, Varadhan SRS (1975) On a variational formula for the principal eigen- dispersal rates: a reaction diffusion model. J Math Biol 37:61–83. value for operators with maximum principle. Proc Nat'l Acad Sci USA 72:780–783. 25. Cantrell R, Cosner C, Lou Y (2010) Evolution of dispersal in heterogeneous landscapes. 62. O’Cinneide C (2000) Markov additive processes and Perron-Frobenius eigenvalue Spatial Ecology, eds R Cantrell, C Cosner, and S Ruan (Chapman & Hall/CRC Press, inequalities. The Annals of Probability 28:1230–1258. London), pp 213–229. 63. Bapat RB, Raghavan TES (1997) Nonnegative Matrices and Applications (Cambridge 26. Hutson V, Martinez S, Mischaikow K, Vickers G (2003) The evolution of dispersal. University Press, Cambridge, United Kingdom). J Math Biol 47:483–517. 64. Schreiber SJ, Lloyd-Smith JO (2009) Invasion dynamics in spatially heterogeneous 27. Lozinskiy S (1969) On an estimate of the spectral radius and spectral abscissa of a environments. Am Nat 174:490–505. matrix. Linear Algebra Appl 2:117–125. 65. Lindqvist BH (2002) On comparison of the Perron-Frobenius eigenvalues of two 28. Deutsch E, Neumann M (1985) On the first and second order derivatives of the Perron ML-matrices. Linear Algebra Appl 353:257–266. vector. Linear Algebra Appl 71:57–76. 66. Caswell H (2000) Matrix Population Models (Sinauer Associates, MA), 2nd Ed., p 722. 29. Keilson J (1964) A review of transient behavior in regular diffusion and birth-death 67. Belgacem F, Cosner C (1995) The effects of dispersal along environmental gradients on processes. J Appl Probab 1:247–266. the dynamics of populations in heterogeneous environments. Canadian Applied 30. Horn RA, Johnson CR (1985) Matrix Analysis (Cambridge University Press, Cambridge). Mathematics Quarterly 3:379–397. 31. Gantmacher F (1959) Applications of the Theory of Matrices (Interscience, NY). 68. Hambrock R (2007) Evolution of conditional dispersal: a reaction-diffusion-advection 32. Seneta E (1981) Non-negative Matrices and Markov Chains (Springer-Verlag, approach. Ph.D. thesis (Ohio State University, OH). New York). 69. Hambrock R, Lou Y (2009) The evolution of conditional dispersal strategies in spatially 33. Bellman R (1955) On an iterative procedure for obtaining the Perron root of a positive heterogeneous habitats. B Math Biol 71:1793–1817. matrix. P Am Math Soc 6:719–725. 70. Altenberg L (2011) An evolutionary reduction principle for mutation rates at multiple 34. Hadeler K, Thieme H (2008) Monotone dependence of the spectral bound on the loci. B Math Biol 73:1227–1270. transition rates in linear compartment models. J Math Biol 57:697–712. 71. Altenberg L (2012) The evolution of dispersal in random environments and the 35. Birindelli I, Mitidieri E, Sweers G (1999) The existence of the principal eigenvalue for principle of partial control, Ecological Monographs, http://arxiv.org/abs/1106.4388. cooperative elliptic systems in a general domain. Diff Equat 35:326–334. 72. Lou Y (2008) Some challenging mathematical problems in evolution of dispersal and 36. Nagel R, ed. (1986) One-parameter Semigroups of Positive Operators, Lecture Notes in population dynamics. Lect Notes Math 171–205. Mathematics (Springer-Verlag, Berlin), 1184. 73. Chen X, Hambrock R, Lou Y (2008) Evolution of conditional dispersal: a reaction- 37. Van Neerven J (1996) The Asymptotic Behaviour of Semigroups of Linear Operators, diffusion-advection model. J Math Biol 57:361–386. Operator Theory Advances and Applications (Birkhäuser, Basel, Switzerland), 88. 74. Bezugly A (2009) Reaction-diffusion-advection models for single and multiple species. 38. Arendt W, Batty C, Hieber M, Neubrander F (2011) Vector-valued Laplace Transforms Ph.D. dissertation (Ohio State University, Columbus, OH). and Cauchy Problems, Monographs in Mathematics, (Birkhäuser, Basel), 2nd Ed., 96. 75. Godoy T, Gossez J, Paczka S (2010) On the asymptotic behavior of the principal eigen- 39. Arendt W (1987) Resolvent positive operators. P Lond Math Soc 3:321–349. values of some elliptic problems. Ann Mat Pur Appl 189:497–521. 40. Arendt W, Batty C, Robinson D (1990) Positive semigroups generated by elliptic 76. Bezuglyy A, Lou Y (2010) Reaction-diffusion models with large advection coefficients. operators on Lie groups. J Operat Theor 23:369–407. Appl Anal 89:983–1004. 41. Ulm M (1999) The interval of resolvent-positivity for the biharmonic operator. PAm 77. Kao C, Lou Y, Shen W (2012) Evolution of mixed dispersal in periodic environments. Math Soc 127:481–490. Discrete and Continuous Dynamical Systems B, 17 In press.

42. Simon B (1982) Schrödinger semigroups. P Am Math Soc 7:447–526. 78. Webb G (1993) Convexity of the growth bound of CO-semigroups of operators. Semi- 43. Voigt J (1986) Absorption semigroups, their generators, and Schrödinger semigroups. groups of Linear Operators and Applications, eds G Goldstein and J Goldstein (Kluwer, J Funct Anal 67:167–205. Dordrecht), pp 259–270.

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