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1 This article was downloaded by:[petrovskii, Sergei V.] On: 29 February 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Biological Dynamics Publication details, including instructions for authors and subscription information: The importance of census times in discrete-time growth-dispersal models Frithjof Lutscher a ; Sergei V. Petrovskii b a Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada b Department of Mathematics, University of Leicester, Leicester, UK Online Publication Date: 01 January 2008 To cite this Article: Lutscher, Frithjof and Petrovskii, Sergei V. (2008) 'The importance of census times in discrete-time growth-dispersal models', Journal of Biological Dynamics, 2:1, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Biological Dynamics Vol. 2, No. 1, January 2008, The importance of census times in discrete-time growth-dispersal models Frithjof Lutscher a and Sergei V. Petrovskii b * a Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada; b Department of Mathematics, University of Leicester, Leicester, UK (Received 07 September 2007; final version received 14 October 2007 ) Dispersal has been the focus of spatial ecology for a few decades. What should be a proper theoretical framework for understanding and modelling of dispersal processes remains a controversial issue though. Integrodifference equations (IDE) model the spatial dynamics of a population with distinct growth and dispersal stages in their life cycle. Depending on the stage observed, the equations take on different forms, only one of which is usually studied in the literature. Here we reveal that while these different forms are mathematically equivalent, the biological conclusions drawn from the different forms may differ considerably. We provide a summary of similarities and differences and point out the greatest potential caveats when applying IDE. Keywords: species dispersal; integrodifference equation; extinction; Allee effect; spatial pattern AMS Subject Classifications: 92B05; 92F05; 39A11; 45M99 1. Introduction A thorough understanding of spatial processes and mechanisms is of crucial importance to explaining and predicting ecosystem dynamics [4]. Many recent studies show that the dynamics predicted by spatial models may differ considerably from those predicted by their non-spatial analogues [6]. The various spatial factors and processes include heterogeneity [13] and long-range interactions [12], but dispersal has undoubtedly received the most attention. The most commonly used theoretical framework for modelling population dynamics in space, which draws on the seminal works by Fisher [11], Skellam [27] and Okubo [21] (see also [2,22]), is based on reaction diffusion equations and is successfully applied in areas such as invasion biology [24,26] and conservation ecology [9]. Diffusion models assume that population growth and dispersal occur on the same time scale. They might therefore be less appropriate to model populations, such as those of certain plants or insects, with clearly distinguished growth and dispersal stages in their life cycle [1]. *Corresponding author. sp237@le.ac.uk ISSN print/issn online 2008 Taylor & Francis DOI: /

3 56 F. Lutscher and S.V. Petrovskii Integrodifference equations (IDE) model these different life stages explicitly and incorporate dispersal via a redistribution kernel [14]. Using various kernels, not necessarily Gaussian, allows one to consider population dynamic consequences of different dispersal behaviour [15 20,29]. Application of IDE models to real-world problems has resulted in new insights and helped to resolve several important ecological problems, for example, in invasion biology [3,7,10,25] or habitat fragmentation [5,8]. When mathematical modelling is used to make real-world predictions, one must ensure that the models are robust and consistent with real-world processes. The additional freedom and flexibility in the IDE modelling framework requires great care when applied to certain situations where the succession of growth and dispersal stages matters for the fate and/or spatial distribution of the species. In this paper, we demonstrate that the same biological process can be described by several mathematically different equations, only one of which is usually studied in the literature. We give two examples of how biological conclusions can be fundamentally different, and potentially wrong, unless the correct formulation is used. We explain how the different formulations are mathematically equivalent, which somehow justifies the treatment of only one type in the literature. This leads us to a summary of the similarities and differences between the different model formulations, and we point out the greatest caveats when applying these models to the real-world problems. 2. The model The life cycle of an organism with separated growth and dispersal stages results in the following succession of stages in the population dynamics (e.g., see [7]): { } { } { } { } adults f dispersers K settlers g adults. (1) generation t generation t generation t generation t + 1 We denote the density of settlers, dispersers, and adults in generation t by u t, ũ t and û t, respectively. Production and settling are given by the functions f and g, respectively, and both are potentially density-dependent processes. We will specify functional forms in the examples below. Dispersal is described by an integral operator with an appropriately chosen dispersal kernel [20]: u t (x) = K[ũ t ](x) := K(x y)ũ t (y)dy. (2) Depending on the stage at which a census is taken, we arrive at one of the following three IDE: (I) u t+1 = K[f(g(u t ))], (II) ũ t+1 = f(g(k[ũ t ])), (3) (III) û t+1 = g(k[f(û t )]). We refer to these models with initial conditions u 0, ũ 0, û 0, as types I, II, and III, respectively. Hence, type I considers the processes of growth and production to act first, whereas type II considers dispersal first. Obviously, if the functions f, g are nonlinear, then the above equations are not identical and thus, properties of the respective solutions might differ significantly. Nonetheless, mathematical analysis has almost exclusively concentrated on the type I model (but see [1]). Intuitively, the behaviour of solutions of these three different equations should be equivalent in some sense, since all three describe the same population life cycle.

4 Journal of Biological Dynamics 57 Before we give a precise definition of how these three equations are equivalent, we give two examples that show that this equivalence is far from obvious. First, we show that for a large class of functions f, we can find initial values for which the solutions of Equation (3-II) converge to zero while solutions of Equation (3-I) grow without bound or to a stable positive equilibrium. In the second example, we find another class of functions f for which a steady-state solution of Equation (3-I) is concave, i.e., it has a single maximum, while the steady state of Equation (3-II) consists of two maxima, i.e., it predicts a different spatial pattern. Finally, we discuss precisely how the three types are equivalent and give a recipe of how incorrect conclusions from differences such as those demonstrated in the examples can be avoided. For the sake of brevity, below we concentrate on the type I and II models to construct examples where solutions are qualitatively different. Consequently, in order to simplify notations we define F (u) = f (g(u)) First example We consider a population on an unbounded domain and choose the growth function to be F (u) = u m, (4) where m>1is a parameter. We choose the -shaped, top-hat dispersal kernel K(z) = 1 for z L, otherwise K(z) = 0, (5) with maximal dispersal distance L. In the appendix, we prove that there is a range of values U 0 > 1 such that from the initial condition u 0 (x) =ũ 0 (x) = U 0 exp( x 2 ), solutions to the type I model grow without bound while solutions to the type II model decay to zero. Here, we give a heuristic idea how growth and dispersal act on the population density to explain why the behaviour of the two solutions is different. The function F in Equation (4) exhibits an Allee effect, i.e., u = 1 is an unstable steady state for the non-spatial equation u t+1 = F(u t ), so that for U 0 <u, the population decreases and for U 0 >u, it increases. This process happens at each point in space. Dispersal acts differently, it tends to smooth out a spatially inhomogeneous distribution by decreasing the value of u at local maxima and increasing it at local minima. Thus, dispersal acts not on the absolute value but on the sharpness of the spatial profile (as given by gradient and curvature). In particular, dispersal has no effect on a spatially homogeneous distribution. Now, let us consider a dome-shaped initial distribution of the population with the maximum value u max. In the case u max <u, both processes act similarly in the sense that they both decrease the value at the maximum. Regardless of whether population growth or dispersal act first, the density decreases to zero in the type I as well as the type II model, i.e., the population goes extinct. However, in the case u max > 1, reproduction and dispersal act in opposite directions. Moreover, while the impact of reproduction depends solely on the value u max, the impact of dispersal depends on the width of the peak. Dispersal decreases the height of a narrow peak considerably, but its impact on a wide peak can be insignificant. For an intermediate value, the impact of dispersal has about the same magnitude as the impact of reproduction, but acting in the opposite direction. This is the case where one can expect that the actual succession of these processes is important, i.e., that the solutions of the type I and type II models may differ essentially. The proof in the appendix is based on the overly simply growth function F in Equation (4) and leads to unbounded population growth. For a more realistic bounded function, such as F (u) = u m (1 + ɛu m ) (6)

5 58 F. Lutscher and S.V. Petrovskii Figure 1. The same initial condition (blue line, t = 0) leads to qualitatively different solutions for the type I equation (green, top panel) and the type II equation (red, bottom panel). Note the different scales on the y-axis. The dispersal kernel is the top-hat kernel (Equation (5)) with variance σ 2 = 0.1 (i.e., L = ), the function F is given by Equation (6) with m = 2 and ɛ = 0.1. (with ɛ>0), populations remain bounded but the main result still holds, namely that the solutions to the type I and type II models differ. We illustrate this case in Figure 1 where, starting from the same initial condition, the population described by model I grows while the population described by model II goes extinct Second example We now consider a bounded domain, l x l, in which the population grows according to the Ricker growth function F (u) = ruexp( βu). (7) We assume that the population disperses according to a Laplace kernel K(z) = a exp( a z ) (8) 2 and that the environment outside the domain is hostile so that, at any t,u t (x) = 0 for x >l. If the non-spatial iteration u t+1 = F(u t ) has a stable positive steady state, then both spatial model types I and II will also have a positive stable steady state provided that the variance of the kernel is small compared to the domain size, i.e., enough individuals are retained within the domain [28]. For the type I model, the spatial shape of the steady state can be approximated by the dispersal success function [18,28], which is convex and symmetric, i.e., has a single maximum in the middle of the domain and decreases as it approaches the boundary (see Appendix for details). For the type II model, however, the shape of the steady-state distribution can have two maxima on the boundary and a trough in the middle of the domain. This is due to the overcompensatory dynamics of the

6 Journal of Biological Dynamics 59 Figure 2. The spatial profile of steady states to model type I (dashed) and type II (solid). The parameters are r = 6, β = 1, and a = 20, i.e., the variance of the kernel is 0.1. Ricker function (7). Similarly to the first example, the two processes of population growth and dispersal act in opposite directions near the boundary: dispersal decreases the density because individuals leave the domain, dynamics increase the density beyond the steady-state level due to overcompensatory growth. These results are illustrated in Figure Discussion and conclusions In the previous section, we showed that the exact form of an IDE describing a population with distinct growth and dispersal stages depends on which of the stages is considered first. We gave two examples that demonstrated that the different mathematical formulations may lead to qualitatively different results or predictions, even though the same processes are considered. The resolution to this apparent paradox comes from a closer inspection of the problem, which takes into account not only the equations but also the corresponding initial conditions. Mathematically speaking, model types I and II are equivalent in the following sense: {u 1,u 2,...} is a solution of the type I model with initial condition u 0 if and only if {F(u 1 ), F (u 2 ),...} is a solution of the type II model with initial condition ũ 0 = F(u 0 ) (see Appendix). This means that solutions are in one-to-one correspondence provided we consider the transformed initial conditions. Similarly, steady states are in one-to-one correspondence (if u (x) is a steady state of the type I equations, then ũ = F(u ) is a steady state of the type II equation and vice versa), and the stability of corresponding steady states is the same for both model types. This definition of equivalence obviously makes sense biologically: if the population is considered in a different stage, then the initial conditions have to be adapted for that stage. In mathematical terms, it provides a certain extension of a concept known as topological conjugacy (see Appendix for details). The transformation of the initial values was not made in the first example, hence leading to qualitatively different predictions for model types I and II. An important observation to be made here is that, if a population experiences an Allee effect, then it is crucial to know whether a given population count refers to the juvenile or adult stage if one wants to predict the fate of the population.

7 60 F. Lutscher and S.V. Petrovskii Similarly, if population growth is density-dependent, then pre- and post-dispersal stages might have considerably different spatial shapes, as was demonstrated by the second example. Interestingly, mathematical issues similar to those discussed above can arise from different biological questions. While we consider how the dynamics depend on the succession of stages in the given life cycle, Andersen [1] studied how the spatial dynamics of a population depends on the timing of density dependence within the life cycle. However, he concluded that the large-time asymptotical behaviour for both models is qualitatively the same. On the contrary, in our work we have identified a case when the models properties are essentially different. IDEs describe populations with different (non-overlapping) life stages, and as such, special care must be taken when formulating, analyzing and parameterizing such models. Typically, the analysis is done with the type I model, which might or might not be the appropriate form. Below we list the potential caveats and highlight the ways around them. If the function F does not have an Allee effect and no overcompensatory growth, then the difference between type I and type II models is small. On an unbounded domain, both predict the existence of a globally attractive positive steady state [30]. On a bounded domain, the conditions on domain size for there to be a globally attracting positive steady state are the same for both equations. In either case, the prediction about persistence or extinction is the same for type I and type II. Only the values at the steady states will differ between the models, but the overall shape of the steady-state solutions will be the same. Therefore, this case is insensitive to small errors in the measurement or the precise shape of F. If the function F exhibits overcompensatory growth but no Allee effect, then the model predictions may differ with respect to the shape of the spatial steady-state distribution as in the second example, but not with respect to stability. If the function F has anallee effect (i.e., for the non-spatial model u t+1 = F(u t ), there exists an unstable threshold population density separating the ranges of population growth and population decline), then things are more complicated. For a realistic parametrization, cf. Equation (6), both the type I and type II model will be bistable. The same initial distribution can converge to zero for the type II model and to the positive steady state in the type I model, but never the other way around. It is in this case when care in model building and parametrization are particularly crucial. The mathematical insights and results above lead to two biological consequences. The first, and most obvious, is that if an integrodifference model is used along with data of field work or laboratory experiments, the choice between types I and II should be made according to and consistent with the origin of the data. The more subtle point is that while the models are mathematically equivalent, their applicability might vary from species to species. While the type I model describes the spatial distribution of individuals after dispersal (such as trees or shrubs or insects in their adult life stage), the type II model describes the spatial distribution of siblings (correspondingly, seeds and eggs or larvae) before dispersal. Some species might naturally lead to a type I model whereas others give rise to a type II model. For example, in annual flowers with windborne seeds, the density of adults is easily accessible while the density of seeds is almost impossible to measure. Vice versa, for certain insects such as the forest tent caterpillar (Malacosoma disstria), egg masses can be collected after hatching and the number of first-instar larvae can be estimated [5], whereas it is almost impossible to count or estimate the number of adult moths. With some species being of type I and others of type II, mathematical results need to be formulated for both types of models and not just for type I, as is usually done in the literature. References [1] M. Andersen, Properties of some density-dependent integrodifference equation population models, Math. Biosci. 104 (1991), pp [2] R.S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, New York, 2003.

8 Journal of Biological Dynamics 61 [3] J.S. Clark et al., Reid s paradox of rapid plant migration, BioSci. 48 (1998), pp [4] J. Clobert et al.(eds.), Dispersal, Oxford University Press, Oxford, [5] C. Cobbold et al., How parasitism affects critical patch size in a host-parasitoid system: application to forest tent caterpillar, Theor. Pop. Biol. 67 (2005), pp [6] R. Durrett and S.A. Levin, The importance of being discrete (and spatial), Theor. Pop. Biol. 46 (1994), pp [7] R. Etienne et al., The interaction between dispersal, the Allee effect and scramble competition affects population dynamics, Ecol. Model. 148 (2002), pp [8] W.F. Fagan and F. Lutscher, Average dispersal success: Linking home range, dispersal, and metapopulation dynamics to reserve design, Ecol. Appl. 16 (2006), pp [9] W.F. Fagan, R.S. Cantrell, and C. Cosner, How habitat edges change species interaction, Am. Nat. 153 (1999), pp [10] W.F. Fagan et al., When can herbivores slow or reverse the spread of an invading plant? A test case from Mount St. Helens, Am. Nat. 166 (2005), pp [11] R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), pp [12] S.A. Gourley and N.F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol. 34 (1996), pp [13] N. Kinezaki et al., Modeling biological invasions into periodically fragmented environments, Theor. Pop. Biol. 64 (2003), pp [14] M. Kot and W.M. Schaffer, Discrete-time growth-dispersal models, Math. Biosci. 80 (1986), pp [15] M. Kot, M.A. Lewis, and P. van der Driessche, Dispersal data and the spread of invading organisms, Ecology 77 (1996), pp [16] M.A. Lewis et al., A guide to calculating discrete-time invasion rates from data, inconceptual Ecology and Invasion Biology: Reciprocal Approaches to Nature, M.W. Cadotte, S.M. McMahon, and T. Fukami, eds., Kluwer, Dordrecht, 2005, pp [17] F. Lutscher, A short note on short distance dispersal, Bull. Math. Biol. 69 (2007), pp [18] F. Lutscher and M.A. Lewis, Spatially-explicit matrix models. A mathematical analysis of stage-structured integrodifference equations, J. Math. Biol. 48 (2004), pp [19] M.G. Neubert and H. Caswell, Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations, Ecology 81 (2000), pp [20] M.G. Neubert, M. Kot, and M.A. Lewis, Dispersal and pattern formation in a discrete-time predator-prey model, Theor. Pop. Biol. 48 (1995), pp [21] A. Okubo, Diffusion and Ecological Problems, Springer, New York, [22] A. Okubo and S.A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, [23] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, [24] S.V. Petrovskii and B.-L. Li, Exactly Solvable Models of Biological Invasion, Chapman & Hall/CRC, Boca Raton, [25] P. Schofield, Spatially explicit models of Turelli Hoffmann Wolbachia invasive wave fronts, J. Theor. Biol. 215 (2002), pp [26] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, [27] J.G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951), pp [28] R.W. Van Kirk and M.A. Lewis, Integrodifference models for persistence in fragmented habitats, Bull. Math. Biol. 59 (1997), pp [29] M.-E. Wang, M. Kot, and M.G. Neubert, Integrodifference equations, Allee effects, and invasions, J. Math. Biol. 44 (2002), pp [30] H.F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), pp Appendix. Calculations and technical details A.1. The first example The type I and type II model equations from Equation (3) can be written as u t+1 (x) = KF [u t (x)] and ũ t+1 (x) = FK[ũ t (x)], (A1) respectively, where F [u](x) = F (u(x)) = f (g(u(x))). For F (u) = u m, the function u(x, t) 1 is an unstable stationary point of F so that for n, F n [u 0 (x)] tends to zero if max (x) u(x, 0) <1 and it grows unboundedly if max (x) u(x, 0) >1. For the kernel given by Equation (5), and indeed for most standard dispersal kernels, the operator K acts similarly to the usual diffusion operator, in that it tends to smooth out the spatial profile of the function. In particular, max (x) K[u(x)] max (x) u(x). Now, assume that we know how the maximum of the solution changes after the first iteration; what can we tell about the properties of the solution in the large-time limit? Taking the above properties into account, we see that, if max (x) u 1 (x) < 1, then max (x) u n (x) 0, whereas if max (x) u 1 (x) > max (x) u 0 (x), then max (x) u n (x). The same conclusion applies to ũ.

9 62 F. Lutscher and S.V. Petrovskii In the case of Equation (3-I) and the initial condition u 0 (x) = U 0 exp( x 2 ), the population distribution after the first iteration will be as follows: u 1 (x) = U0 m 1 x+l e my2 dy x L = U0 m 1 1 π m 2 ( [ m(x + L)] [ m(x L)]), (A2) where (z) = (2/ π) z 0 exp( x2 )dx is the error function. Obviously, the maximum is reached at x = 0 so that max u 1 (x) = u 1 (0) = U m ( 0 π ) 1/2 ( ml) α. (A3) x m In the case of Equation (3-II), the population density after the first iteration is given as ( U0 ũ 1 (x) = x+l m e dy) y2, (A4) x L where U x+l 0 e y2 dy = U 0 π [ (x + L) (x L)]. (A5) x L 2 Correspondingly, the maximum population density is max ũ 1 (x) =ũ 1 (0) = β m, (A6) x where β = U 0 π (L). (A7) We claim that we can find values U 0 and L for which α>u 0 and β<1. Due to the properties of F and K, this implies that, starting from the same initial condition, in the large-time limit, the solution of Equation (3-II) tends to zero while the solution of Equation (3-I) grows unboundedly. From Equations (A3) and (A7), we arrive at the system of the following two relations: (a) U m 1 0 ( π m ) 1/2 ( ml) > 1 (A8) (b) U 0 π (L) < 1. There are different ways to see that the solution of the system (A8) is not an empty set, provided m is sufficiently large. One of them is shown below. For any fixed value of L relation (A8(b)) gives, in the case when inequality changes to equality, a critical value u, u = 2 L φ (L), (A9) so that for any U 0 <u, the solution of Equation (3-II) with the given initial condition will tend to zero. Note that, taking into account the properties of the error function, it is readily seen that u > 1 for any L>0. Similarly, from Equation (A8(a)), we obtain a critical value u, [ ] 1/(m 1) 2 ml u = π (, (A10) ml) so that for any U 0 >u, the solution of Equation (3-I) will grow unboundedly. In order to prove consistency of relations (A8), it is sufficient to prove that u >u. Then, for any U 0 with u < U 0 <u, the solution of Equation (3-I) grows unboundedly while the solution of Equation (3-II) decays to zero. A direct comparison between Equations (A9) and (A10) seems to be very difficult, though. Instead, we can show that, for some m and L, inequality (A8(a)) remains true when U 0 = u. Indeed, having substituted U 0 = u into (A8(a)), we obtain: u m 1 m ( ml) > π. (A11) Since u m 1 / m for m (recall that u > 1) and (z) 1 for z, it is obvious that the inequality holds if m is sufficiently large. That completes the proof.

10 Journal of Biological Dynamics 63 A.2. The second example For a (symmetric) dispersal kernel K(z) = K( x y ), the dispersal success function on the domain [ l,l]isgivenby l s(x) = K( x y )dy, (A12) l see [28] for details. It gives the probability that an individual that disperses from x remains in the domain. Since individuals near the domain boundary are more likely to leave the domain than individuals in the middle of the domain, this function has its maximum at the centre of the domain and decreases towards the boundary [28]. Expanding the steady-state equation for model type I in Taylor series, and keeping only the lowest terms, one gets U(x) F(Ū)s(x), (A13) where U is the steady-state density and Ū its spatial average that is obtained from the equation Ū = F(Ū) s, (A14) with s, the spatial average of s(x) on [ l; l]. Hence, the steady-state profile of U has approximately the same shape as the dispersal success function. If we apply Taylor series expansion to the type II model, we obtain U(x) (Ūs(x)). (A15) If F exhibits overcompensatory growth then the smaller values of s near the boundary become large values of U whereas larger values of s in the middle of the domain are decreased or at least not increased as much as the values near the boundary. A.3. Equivalence of the three types of equations Assuming that the sequence u 1 (x), u 2 (x),...is a solution for the type I model, we define a new sequence v t (x) = g(u t (x)). Then we see that v t+1 (x) = g(u t+1 (x)) = g(k[f(g(u t (x)))]) = g(k[f(v t (x))]). (A16) Hence, the sequence v t (x) satisfies the type III model equation, and therefore has to equal the solution û t (x) with the initial condition û 0 (x) = v 0 (x) = g(u 0 (x)). Therefore, the three model equations are equivalent by a shift using the functions f, g or K, respectively. This equivalence requires that the initial value be transformed as well. Equivalence for the type I and II models, in a different context, was discussed earlier in [1]. In more general mathematical terms, two dynamical systems u t+1 = D 1 (u t ) and v t+1 = D 2 (v t ) are called topologically conjugate if there is a homoemorphism H such that H(D 1 (u)) = D 2 (H (u)) for all u (e.g., see [23]). The type I model is given by D 1 = KF whereas the type II model is given by D 2 = FK. IfwesetH = F, then the conjugacy condition is satisfied. Similarly, if we set H = K and interchange the indices. However, in general neither F nor K are invertible, and hence the map H is not a homoemorphism. Exceptions are the first example or the Beverton Holt function for which the map H is invertible. Future mathematical work will have to investigate a weaker definition of topological conjugacy that captures the situation here, and explore its consequences.