Coagulation–Fragmentation Model for Animal GroupSize Statistics
Abstract
We study coagulation–fragmentation equations inspired by a simple model proposed in fisheries science to explain data for the size distribution of schools of pelagic fish. Although the equations lack detailed balance and admit no Htheorem, we are able to develop a rather complete description of equilibrium profiles and largetime behavior, based on recent developments in complex function theory for Bernstein and Pick functions. In the largepopulation continuum limit, a scalinginvariant regime is reached in which all equilibria are determined by a single scaling profile. This universal profile exhibits powerlaw behavior crossing over from exponent \(\frac{2}{3}\) for small size to \(\frac{3}{2}\) for large size, with an exponential cutoff.
Keywords
Detailed balance Fish schools Bernstein functions Complete monotonicity Fuss–Catalan sequences Convergence to equilibriumMathematics Subject Classification
45J05 70F45 92D50 37L15 44A10 35Q991 Introduction
A variety of methods have been used to account for the observed statistics of animal group size in population ecology. We refer to the works of Okubo (1986), Okubo and Levin (2001), and Couzin and Krause (2003) for extensive discussions of dynamical aspects of how animal groups form and evolve, including quantitative approaches. The present work focuses on coagulation–fragmentation equations, which model rates of merging and splitting of groups. These kinds of models seek to account for the observed frequency distribution of group sizes by simple rules involving the fission and fusion rates, which subsume all details of individual behavior and internal group structure.
The models that we study are particularly motivated by several works aimed at explaining observations in fisheries science that concern the size distribution of schools of pelagic fish, which roam in the midocean (Bonabeau and Dagorn 1995; Bonabeau et al. 1998; Niwa 1996, 1998, 2003, 2004; Ma et al. 2011). For such creatures, it is plausible that groupsize dynamics may be modeled as spatially homogeneous and dominated by random encounters. These and related works have also discussed data for other kinds of animal groups, including mammalian herds and flocks of geese (Hayakawa and Furuhashi 2012) and sparrows (Griesser et al. 2011).

The ocean is modeled as a discrete set of sites that fish schools may occupy.

Schools jump to a randomly chosen site at discrete time steps.

Two schools arriving at the same site merge.
 Any school may split in two with fixed probability per time step,

independent of the school size i,
 with uniform likelihood among the \(i1\) splitting outcomes$$\begin{aligned} (1,i1), (2,i2),\ldots ,(i1,1). \end{aligned}$$

The reason for this dearth of theory is that the existing results that concern equilibria and longtime behavior almost all deal with systems that admit equilibria in detailed balance, with equal rates of merging and splitting for each reaction taking clusters of sizes i, j to one of size \(i+j\). For modeling animal group sizes in particular, Gueron and Levin (1995) discussed several coagulation–fragmentation models for continuoussize distributions with explicit formulae for equilibria having detailed balance. However, Niwa argued explicitly in Niwa (2003) that the observational data for pelagic fish are inconsistent with these models.
Our purpose in the present paper is to carry out a mathematically rigorous investigation of coagulation–fragmentation equations motivated by Niwa’s model and the work (Ma et al. 2011). For a continuoussize model, and for a discretesize model that differs only slightly from that of Ma et al. (2011), we will describe the equilibria in detail, show that these solutions globally attract all solutions with finite total population, and establish convergence in the discretetocontinuum limit.
This characterization of equilibria in terms of the profile \(\Phi _{\star }\) confirms Niwa’s finding of scale invariance, as a rigorous consequence of coagulation–fragmentation modeling in the continuum limit. The powerlaw behavior exhibited by the exponential prefactor g(x) changes as one goes from small to large group sizes, however, from \(x^{2/3}\) for small x to \(x^{3/2}\) for large x. This suggests that it could be difficult in practice to distinguish the profile in (1.5) from the expression in (1.2) with prefactor \(x^{1}\).
Indeed, in Fig. 2 we compare Niwa’s profile in (1.2) to the new profile \(\Phi _{\star }\) in (1.5) as computed using 45 terms from the desingularized power series in Sect. 5.4, and (5.7). The scaling of the log–log plot facilitates comparison with the scaled empirical data in Fig. 1 above (Niwa 2003, Figure 5), in which the solid line plots the profile (1.2). It is evident from this comparison that the new profile \(\Phi _{\star }\) fits this data essentially as well as (1.2).
Dynamics without detailed balance. In addition to this description of equilibrium, we develop a rather complete theory of dynamics in the continuum limit, for weak solutions whose initial data are finite measures on \((0,\infty )\). We establish convergence to equilibrium for all solutions that correspond to finite total population (finite first moment). Furthermore, for initial data with infinite first moment, solutions converge to zero in a weak sense, meaning that the population concentrates in clusters whose size grows without bound as \(t\rightarrow \infty \).
Previous mathematical studies of equilibria and dynamical behavior in coagulation–fragmentation models include work of Aizenman and Bak (1979), Carr (1992), Carr and Costa (1994), Laurençot and Mischler (2003), and Cañizo (2007), as well as a substantial literature related to Becker–Doering equations (which take into account only clustering events that involve the gain or loss of a single individual). These works all concern models having detailed balance and rely on some form of H theorem, or entropy/entropydissipation arguments. For models without detailed balance, there is work of Fournier and Mischler (2004), concerning initial data near equilibrium in discrete systems, and a recent study by Laurençot and Roessel (2015) of a model with a multiplicative coagulation rate kernel in critical balance with a fragmentation mechanism that produces an infinite number of fragments.
Arguments involving entropy are not available for the models that we need to treat here. Instead, it turns out to be possible to use methods from complex function theory related to the Laplace transform. Such methods have been used to great advantage to analyze the dynamics of pure coagulation equations with special rate kernels (Menon and Pego 2004, 2008; Laurençot and Roessel 2010, 2015).
Specifically what is relevant for the present work is the theory of Bernstein functions, as developed in the book of Schilling et al. (2010). In terms of the “Bernstein transform,” the coagulation–fragmentation equation in the continuum limit transforms to a nonlocal integrodifferential equation which turns out to permit a detailed analysis of equilibria and longtime dynamics.
In particular, for this discretesize model, we can characterize its equilibrium distributions (which depend now on total population) in terms of completely monotone sequences with exponential cutoff. Whenever the total population is initially finite, every solution converges to equilibrium strongly with respect to a sizeweighted norm. And for infinite total population, again the population concentrates in everlarger clusters—the size distribution converges to zero pointwise while the zeroth moment goes to a nonzero constant.
Plan. The plan of the paper is as follows. In Sect. 2, we describe the coagulation–fragmentation models under study in both the discretesize setting (Model D) and continuoussize setting (Model C). A summary of results from the theory of Bernstein functions appears in Sect. 3. Our analysis of Model C is carried out in sections 4–9 (Part I), and Model D is treated in sections 10–13 (Part II). Lastly, in Part III (sections 14–16) we relate Model D to a discretization of Model C and prove a discretetocontinuum limit theorem.
2 Coagulation–Fragmentation Models D and C
In this section, we describe the coagulation–fragmentation meanfield rate equations that model Niwa’s merging–splitting simulations, and the corresponding equations for both discretesize and continuoussize distributions that we focus upon in this paper.
2.1 DiscreteSize Distributions
2.2 ContinuousSize Distributions
2.3 Scaling Relations for Models D and C
By simple scalings, we can relate the solutions of Models D and C to solutions of the same models with conveniently chosen coefficients.
3 Bernstein Functions and Transforms
Throughout this paper, we make use of various results from the theory of Bernstein functions, as laid out in the book (Schilling et al. 2010). We summarize here a number of key properties of Bernstein functions that we need in the sequel.
Recall that a function \(g:(0,\infty )\rightarrow {\mathbb {R}}\) is completely monotone if it is infinitely differentiable and its derivatives satisfy \((1)^ng^{(n)}(x)\ge 0\) for all real \(x>0\) and integer \(n\ge 0\). By Bernstein’s theorem, g is completely monotone if and only if it is the Laplace transform of some (Radon) measure on \([0,\infty )\).
Definition 3.1
A function \(U:(0,\infty )\rightarrow {\mathbb {R}}\) is a Bernstein function if it is infinitely differentiable, nonnegative, and its derivative \(U'\) is completely monotone.
The main representation theorem for these functions (Schilling et al. 2010, Thm. 3.2) associates to each Bernstein function U a unique Lévy triple \((a_0,a_\infty ,F)\) as follows. (Below, the notation \(a\wedge b\) means the minimum of a and b.)
Theorem 3.2
Definition 3.3
Whenever (3.1) holds, we call U the Bernstein transform of the Lévy triple \((a_0,a_\infty ,F)\). If \(a_0=a_\infty =0\), we call U the Bernstein transform of the Lévy measure F, and write \(U=\breve{F}\).
Many basic properties of Bernstein functions follow from Laplace transform theory. Yet Bernstein functions have beautiful and distinctive properties that are worth delineating separately. The second statement in the following proposition is proved in Lemma 2.3 of Iyer et al. (2015).
Proposition 3.4
The composition of any two Bernstein functions is Bernstein. Moreover, if \(V:(0,\infty )\rightarrow (0,\infty )\) is bijective and \(V'\) is Bernstein, then the inverse function \(V^{1}\) is Bernstein.
Topologies. Any pointwise limit of a sequence of Bernstein functions is Bernstein. The topology of this pointwise convergence corresponds to a notion of weak convergence related to the associated Lévy triples in a way that is not fully characterized in Schilling et al. (2010), but may be described as follows. Let \({\mathcal {M}}_+[0,\infty ]\) denote the space of nonnegative finite (Radon) measures on the compactified halfline \([0,\infty ]\).
Theorem 3.5
 (i)
\(U(s):=\lim _{n\rightarrow \infty }U_n(s)\) exists for each \(s\in (0,\infty )\).
 (ii)
\(\kappa _n \xrightarrow {w}\kappa \) as \(n\rightarrow \infty \), where \(\kappa \) is a finite measure on \([0,\infty ]\).
This result is essentially a restatement of Theorem 3.1 in Menon and Pego (2008), where different terminology is used. A simpler, direct proof will appear in Iyer et al. (in preparation), however.
Proposition 3.6
 (i)
\(F_n\) converges narrowly to F, i.e., \(F_n\xrightarrow {n}F\).
 (ii)
The Laplace transforms \({\mathcal {L}}F_n(s)\rightarrow {\mathcal {L}}F(s)\), for each \(s\in [0,\infty )\).
 (iii)
The Bernstein transforms \(\breve{F}_n(s) \rightarrow \breve{F}(s)\), for each \(s\in [0,\infty ]\).
 (iv)
The Bernstein transforms \(\breve{F}_n(s) \rightarrow \breve{F}(s)\), uniformly for \(s\in (0,\infty )\).
Proof
Complete monotonicity. Our analysis of the equilibria of Model C relies on a striking result from the theory of the socalled complete Bernstein functions, as developed in Schilling et al. (2010, Chap. 6):
Theorem 3.7
 (i)The Lévy measure F in (3.1) has a completely monotone density g, so that$$\begin{aligned} U(s) = a_0s+a_\infty +\int _{(0,\infty )} (1\hbox {e}^{sx}) g(x)\,\hbox {d}x, \quad s\in (0,\infty ). \end{aligned}$$(3.7)
 (ii)
U is a Bernstein function that admits a holomorphic extension to the cut plane \({\mathbb {C}}{\setminus }(\infty ,0]\) satisfying \((\mathrm{Im}s)\mathrm{Im}U(s) \ge 0\).
In complex function theory, a function holomorphic on the upper half of the complex plane that leaves it invariant is called a Pick function or NevalinnaPick function. Condition (ii) of the theorem above says simply that U is a Pick function analytic and nonnegative on \((0,\infty )\). Such functions are called complete Bernstein functions in Schilling et al. (2010).
Theorem 3.8
 (i)
The sequence \((c_j\lambda ^j)\) is completely monotone.
 (ii)
G is a Pick function that is analytic and nonnegative on \((\infty ,\lambda )\).
 (Part I)

Analysis of Model C
4 Equations for the ContinuousSize Model
In successive sections to follow, we study the existence and uniqueness of weak solutions, characterize the equilibrium solutions, and study the longtime behavior of solutions for both finite and infinite total population size.
5 Equilibrium Profiles for Model C
Theorem 5.1
Remark 5.1
Remark 5.2
5.1 The Bernstein Transform at Equilibrium
Remark
5.2 Relation to Fuss–Catalan numbers
Lemma 5.2
The function \(B_3\) is a Pick function analytic on \({\mathbb {C}}{\setminus }\left[ \frac{4}{27},\infty \right) \) and nonnegative on the real interval \(\left( \infty ,\frac{4}{27}\right) \).
5.3 Proof of Theorem 5.1
5.4 Series Expansion for the Equilibrium Profile
6 Global Existence and Mass Conservation
Next, we deal with the initialvalue problem for Model C. Although this can be studied using rather standard techniques from kinetic theory, we find it convenient to study this problem using Bernstein function theory.
We require solutions take values in the space \({\mathcal {M}}_+(0,\infty )\), which we recall to be the space of nonnegative finite measures on \((0,\infty )\). The main result of this section is the following.
Theorem 6.1
 (i)If the first moment \(m_{1}(F_{\mathrm{in}})<\infty \), thenfor all \(t\in [0,\infty )\), and \(t\mapsto x\,F_t(\hbox {d}x)\in {\mathcal {M}}_+(0,\infty )\) is narrowly continuous.$$\begin{aligned} m_1(F_t) = m_1(F_{\mathrm{in}}) \end{aligned}$$(6.1)
 (ii)
If \(F_{\mathrm{in}}(x)= f_{\mathrm{in}}(x)\,\hbox {d}x\) where \(f_{\mathrm{in}}\) is a completely monotone function, then for any \(t \ge 0\), \(F_{t}(\hbox {d}x)= f_{t}(x)\,\hbox {d}x\) where \(f_{t}\) is also a completely monotone function.
Proof
We may now deduce (4.7) by taking \(s\rightarrow \infty \) in (6.9). (Indeed, because U(s, t) increases as \(s\rightarrow \infty \) toward \(m_0(F_t)\), one finds \(s^{1}\int _0^s U(r,\tau )\,\hbox {d}r\rightarrow m_0(F_\tau )\) for each \(\tau >0\).) Because \(t\mapsto \breve{F}_t(s)\) is continuous for each \(s\in [0,\infty ]\), we may now invoke Proposition 3.6 to conclude that \(t\mapsto F_t\) is narrowly continuous.
At this point, we know that (4.1) holds for each test function of the form \(\varphi (x)=1\hbox {e}^{sx}\), \(s\in (0,\infty ]\). Linear combinations of these functions are dense in the space of continuous functions on \([0,\infty ]\). (This follows by using the homeomorphism \([0,\infty ]\rightarrow [0,1]\) given by \(x\mapsto \hbox {e}^{x}\) together with the Weierstrass approximation theorem.) Then, because \(m_0(F_t)\) is uniformly bounded, it is clear that one obtains (4.1) for arbitrary continuous \(\varphi \) on \([0,\infty ]\) by approximation.
Because \(t\mapsto (x\wedge 1)F_t\) is weakly continuous on \([0,\infty ]\) by the continuity theorem 3.5, and \(m_1(F_t)\) is bounded, the finite measure \(x\,F_t(\hbox {d}x)\) is vaguely continuous, hence it is narrowly continuous because \(m_1(F_t)\) remains constant in time.
Finally we prove (ii). The hypothesis implies that \(U_0\) is a complete Bernstein function by Theorem 3.7. Complete Bernstein functions form a convex cone that is closed with respect to pointwise limits and composition, according to Schilling et al. (2010, Cor. 7.6).
We prove by induction that \(U_n\) is a complete Bernstein function for every \(n\ge 0\). For if \(U_n\) has this property, then clearly \(\hat{U}_n\) does. From the formula (6.5), \(G_{\Delta t}^{1}\) is a complete Bernstein function, as it is positive on \((0,\infty )\), analytic on \({\mathbb {C}}{\setminus }(\infty ,0]\) and leaves the upper half plane invariant. Therefore, \(U_{n+1}\) is completely Bernstein.
Passing to the limit as \(\Delta t\rightarrow 0\), we deduce that \(U(\cdot ,t)\) is completely Bernstein for each \(t\ge 0\). By the representation theorem 3.7, we deduce that the measure \(F_t\) has a completely monotone density as stated in (ii).
This completes the proof of Theorem 6.1, except for uniqueness. Uniqueness is proved by a Gronwall argument very similar to that used below in Sect. 16 to study the discretetocontinuum limit. We refer to Eqs. (16.1)–(16.5) and omit further details. \(\square \)
7 Convergence to Equilibrium with Finite First Moment
In this section, we prove that any solution with finite first moment converges to equilibrium in a weak sense. Our main result is the following.
Theorem 7.1
Proposition 7.2
Proof
Step 2: Upper barrier. Our proof of convergence is based on comparison principles, which will be established with the aid of two lemmas.
Lemma 7.3
Proof of Lemma 7.3
Lemma 7.4
For all \(s\in (0,\bar{s})\) and \(t>0\), \(w(s,t)\le \bar{w}(s,t)\).
Proof of Lemma 7.4
Now, because \(\underline{w}(s,t)\le w(s,t)\le \bar{w}(s,t)\) and \(\underline{w}(s,t)\), \(\bar{w}(s,t)\rightarrow 0\) as \(t\rightarrow \infty \) for each \(s>0\), the convergence result for U(s, t) in the Proposition follows. This finishes the proof of the Proposition. \(\square \)
Proof of Theorem 7.1
The result of the Proposition applies to the Bernstein transform \(U(s,t)=\breve{F}_t(s)\) for arbitrary \(\bar{s}>0\), and this yields (7.3). This finishes the main step of the proof, and it remains to deduce (7.1) and (7.2). We know that \(m_0(F_t)=U(\infty ,t)\rightarrow 1=U_\star (\infty )\) as \(t\rightarrow \infty \), hence by Proposition 3.6 it follows \(F_t(\hbox {d}x)\rightarrow f_\star (x)\,\hbox {d}x\) narrowly as \(t\rightarrow \infty \).
By consequence, because \(m_1(F_t)\equiv 1\) is bounded, the measures \(xF_t(\hbox {d}x)\) converge to \(xf_\star (x)\,\hbox {d}x\) vaguely, and because \(m_1(F_t)\equiv m_1(f_\star )\), this convergence also holds narrowly. This finishes the proof of the Theorem. \(\square \)
8 Weak Convergence to Zero with Infinite First Moment
Next, we study the case with infinite first moment. If the initial data have infinite first moment, the solution converges to zero in a weak sense, with all clusters growing asymptotically to infinite size, loosely speaking.
Theorem 8.1
Proof
9 No Detailed Balance for Model C
Theorem 9.1
For Model C, no finite measure \(F_*\) on \((0,\infty )\) exists that satisfies condition (9.2) for detailed balance in weak form.
Proof
Remark 9.1
 (Part II)

Analysis of Model D
10 Equations for the DiscreteSize Model
Remark 10.1
11 Equilibrium Profiles for Model D
Theorem 11.1
Proof
Asymptotics. The decay rate of the sequence f shall be deduced from the derivative of G(z) using Tauberian arguments, as developed in the book of Flajolet and Sedgewick (2009).
Recall that \(U_\star (s)\) has a branch point at \(s=\frac{4}{27}\), with \(U_\star '\left( s\frac{4}{27}\right) \sim \frac{9}{8} s^{1/2}\) due to (5.26). The generating function G(z) has a corresponding branch point at \(z=\lambda _\mu \).
11.1 Recursive Computation of Equilibria for Model D
12 Well Posedness for Model D
Here, we consider the discrete dynamics described by Eqs. (10.1)–(10.4), and establish well posedness of the initialvalue problem by a simple strategy of proving local Lipschitz estimates on an appropriate Banach space.
Theorem 12.1
\(m_1(f(t))=m_1(f_{\mathrm{in}})\) for all \(t\ge 0\).
Proof
To obtain (12.1), take \(\varphi _i\equiv 1\) in (10.1), or take \(s\rightarrow \infty \) in (10.6). \(\square \)
13 LongTime Behavior for Model D
By using much of the same analysis as in the continuoussize case, we obtain strong convergence to equilibrium for solutions with a finite first moment, and weak convergence to zero for solutions with infinite first moment.
Theorem 13.1
Proof
Theorem 13.2
Remark 13.1
The conclusion means that the total number of groups \(m_0(f(t))\rightarrow 1\), while the number of groups of any fixed size i tends to zero. Thus as time increases, individuals cluster in larger and larger groups, leaving no groups of finite size in the largetime limit.
Proof
 (Part III)

From Discrete to Continuous Size
14 Discretization of Model C
15 Limit Relations at Equilibrium
We have the following rigorous convergence theorem for the continuum limit of the discrete equilibria.
Theorem 15.1
Proof
16 DiscretetoContinuum Limit
We can rigorously prove a weakconvergence result for timedependent solutions of Model D to solutions of Model C, as follows.
Theorem 16.1
Let F be a solution of Model C with initial data \(F_0\) a finite measure on \((0,\infty )\), and let \(f^h\), \(h\in I\) be solutions of Model D with initial data that satisfy \(F^h_0\rightarrow F_0\) narrowly as \(h\rightarrow 0\). Then, for each \(t>0\), we have \(F^h_t\rightarrow F_t\) narrowly as \(h\rightarrow 0\).
Proof of the Theorem
1. The Bernstein functions \(U^h=\breve{F}^h_t\) and \(U=\breve{F}_t\) satisfy (14.6) and (4.4) respectively. For h small enough, these functions are uniformly bounded globally in time by \(C_0=m_0(F_0)+2\), due to the fact that \(m_0(F_0^h)\rightarrow m_0(F_0)\) and the bounds coming from (4.7) and (12.2).
The narrow convergence \(F^h_t\rightarrow F_t\) now follows from Proposition 3.6.\(\square \)
Remark 16.1
Uniqueness for Model C is proved by a simple Gronwall estimate analogous to (16.5). We omit details.
17 Discussion and Conclusions
Niwa (2003) simulated random merging and splitting dynamics according to the rules we listed in the introduction, but he did so for an indirect purpose: The simulations were used as a tool in order to estimate noise terms in a stochastic differential equation that he formulated to model the group size experienced by an individual. He then solved this SDE to obtain selfconsistently the equilibrium group size in the form of his profile (1.2).
The present paper is a study of meanfield, deterministic coagulation–fragmentation equations that closely correspond to Niwa’s simulations. As stochastic effects can be expected to become nontrivial as populations decrease, it would be interesting to study stochastic models in a manner compatible with interaction rules and the deterministic limit. Niwa’s individualbased point of view could lead to a kind of model different from the Markus–Lushnikov merging processes whose largepopulation limits are known rigorously to converge to deterministic coagulation equations (Norris 1999). Because one can expect size of the group containing a given individual to change discontinuously upon mergers, appropriate selfconsistent individualbased models should involve jump processes rather than SDEs.
Consistent with Niwa’s major finding, we have shown that in the continuum limit corresponding to large population size, the equilibrium groupsize distribution does achieve a scalinginvariant form. Moreover, the exact equilibrium profile \(\Phi _{\star }\) in the continuum limit is computable in terms of explicit series representations as described in Sect. 5.4. It is not as simple as Niwa’s expression (1.2), but we have determined its asymptotic form as an exponential with a smooth prefactor having different powerlaw behavior at small vs. large group sizes.
Much of the empirical data exhibited by Niwa (2003) is in a range where the profile \(\Phi _{\star }\) differs little from (1.2) or from the logarithmic distribution (1.9). In principle, it would be interesting to determine whether the profile \(\Phi _{\star }\) is a better model for real data than other alternatives, but it may not be easy given the difficulties of dealing with noisy population data. In particular, the crossover that we found in the powerlaw behavior of the exponential prefactor suggests that extracting powerlaw exponents from data, as is done in a number of papers in the literature including Bonabeau et al. (1999) and Sjöberg et al. (2000), may be difficult to do accurately and reliably.
In broader terms, however, what our study does support is the idea that indeed this kind of widely dispersed, nonGaussian groupsize distribution can arise as an emergent property in a mathematically wellformulated model of random encounters between groups and having a simple kind of group instability.
In our investigation of dynamical behavior, we have proved that for a given finite total population, the unique equilibrium globally attracts all solutions, both for continuoussize and discretesize variants. Because of the rigidity of the techniques that we use based on Bernstein transform, this finding does not extend further, even to models that have the same equilibria but differing rates as indicated in Remark 5.2. More information about how solutions approach equilibrium would be interesting to obtain. Some numerical investigations related to equilibria and dynamical stability for variant discretizations of our continuoussize Model C have been performed in separate work (Degond and Engel 2016).
The systems that we study lack detailed balance, which means that their equilibria are maintained by a steady cycling in the network of reactions between groups of different sizes. Remarkably little is known about dynamics in such systems. We wonder, for example, whether any such coagulation–fragmentation systems may have dynamically cycling (timeperiodic) solutions.
Notes
Acknowledgments
This material is based upon work supported by the National Science Foundation under Grants DMS 1211161, DMS 1515400, and DMS 1514826, and partially supported by the Center for Nonlinear Analysis (CNA) under National Science Foundation PIRE Grant no. OISE0967140, and the NSF Research Network Grant no. RNMS1107444 (KINet). PD acknowledges support from EPSRC under Grant ref: EP/M006883/1, from the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award. PD is on leave from CNRS, Institut de Mathématiques de Toulouse, France. JGL and RLP acknowledge support from the Institut de Mathématiques, Université Paul Sabatier, Toulouse and the Department of Mathematics, Imperial College London, under Nelder Fellowship awards.
Data statement No new data was collected in the course of this research.
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