Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities

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Ecological Complexity journal homepage: www.elsevier.com/locate/ecocom

Original Research Article

Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities$ Ozgur Aydogmusa,* , Yun Kangb , Musa Emre Kavgacic , Huseyin Bereketogluc a

Department of Economics, Social Sciences University of Ankara, Oran, Ankara, Turkey Sciences and Mathematics Faculty, College of Integrative Sciences and Arts, Arizona State University Mesa, AZ 85212, USA c Department of Mathematics, Ankara University, Tandogan, Ankara, Turkey b

A R T I C L E I N F O

Article history: Received 2 January 2017 Received in revised form 22 March 2017 Accepted 9 April 2017 Available online xxx Keywords: Integro-difference equation Nonlocal interaction Pattern formation Multiscale perturbation

A B S T R A C T

The paper is devoted to the study of discrete time and continuous space models with nonlocal resource competition and periodic boundary conditions. We consider generalizations of logistic and Ricker's equations as intraspecific resource competition models with symmetric nonlocal dispersal and interaction terms. Both interaction and dispersal are modeled using convolution integrals, each of which has a parameter describing the range of nonlocality. It is shown that the spatially homogeneous equilibrium of these models becomes unstable for some kernel functions and parameter values by performing a linear stability analysis. To be able to further analyze the behavior of solutions to the models near the stability boundary, weakly nonlinear analysis, a well-known method for continuous time systems, is employed. We obtain Stuart–Landau type equations and give their parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities of two symmetric kernel functions. Our calculations indicate that supercritical bifurcations occur near stability boundary for uniform kernel functions. We also verify these results numerically for both models. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Diffusion driven instabilities in reaction-diffusion equations (RDE) have been studied since Turing's seminal work (Turing, 1952) where he showed that diffusion combined with intra- and inter-specific interactions can lead to instability of space homogeneous equilibrium, and thus generate pattern formation. Segel and Stoeckly (1972) adopted the idea of diffusive instability to the ecology context. Since then there is an enormous literature on pattern formation in ecology (see e.g. Okubo and Levin, 2013; Murray, 2003). Another important mechanism that generates pattern formation in continuous time systems is nonlocal interactions. The effect of such interactions has been studied in Britton (1989, 1990) and Gourley (2000). It was shown that the solutions to nonlocal Fisher equation exhibit instabilities of the space homogeneous solution

$

Fully documented templates are available in the elsarticle package on CTAN. * Corresponding author. E-mail addresses: [email protected] (O. Aydogmus), [email protected] (Y. Kang), [email protected] (M.E. Kavgaci), [email protected] (H. Bereketoglu).

(Genieys et al., 2006, 2009; Perthame and Génieys, 2007; Fuentes et al., 2003, 2004; Banerjee et al., 2017). Their model assumes that species disperses and at the same time competes for resources where the competition term is modeled via a contact distribution and diffusion can be local or nonlocal (Genieys et al., 2006; Aydogmus, 2015). Recently, the effects of nonlocal competition or interaction have been investigated for nonlocal RDEs (Segal et al., 2013; Tanzy et al., 2013; Banerjee and Volpert, 2016a,b). The study of Neubert et al. (1995) suggests that RDEs are not appropriate modeling tools for a large number of species that have discrete non-overlapping generations. Examples of such species consist of some plants (e.g., annual grass) and insects (e.g., paper wasps). Annual dispersal distances for these species are given by Neubert et al. (1995). In literature, such species have been modeled via integro-difference equations and analyzed in terms of traveling waves (Kot, 1992), spread and invasion (Kot et al., 1996), and pattern formation studied by Neubert et al. (1995) in an infinite domain. They considered integro-difference equations as discretetime spatial contact models and showed that solutions to such systems exhibit dispersal driven instabilities. As noted by Britton (1989), intra and interspecific growth terms (i.e. reaction terms) of RDEs depend only on the local population

http://dx.doi.org/10.1016/j.ecocom.2017.04.001 1476-945X/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: O. Aydogmus, et al., Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities, Ecol. Complex. (2017), http://dx.doi.org/10.1016/j.ecocom.2017.04.001

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density. Similarly, if we use integro-difference equations as a modeling tool, we need to assume that the growth term depends on the local population density as well. However for mobile animals having non-overlapping generations such as paper wasps, the important factor for the growth term is the depletion of the common resource in their neighborhood. Therefore, the first aim of this paper is to study the effects of nonlocal interactions in discrete-time logistic and Ricker's equations obtained by adding spatial effects (both dispersal and nonlocal competition for resources) to these models. We perform linear stability analysis of these models with periodic boundary conditions and show that the nonlocal interaction term with certain properties can destabilize the spatially homogeneous equilibrium. The linear stability analysis that we perform on two models is generally used to find pattern formation conditions, and the related technique is local in principle which could help us determine the value of the unstable wavenumbers and the parameters for which the space homogeneous solution becomes unstable. The classical Stuart–Landau (SL) theory (Stuart, 1960), on the other hand, widely applied to RDE's in biology (Murray, 2003), has been employed for the further investigation of the complex structure of the attractors. This method is also applied to nonlocal aggregation models (Topaz et al., 2006; Eftimie et al., 2009) and it was reported that the transition to instability is subcritical i.e. near the stability boundary, one can observe large oscillations depending on the initial condition. In other words, transition from disordered to ordered behavior is discontinuous. As a note, this method was also used in a recent work (Aydogmus, 2015) for the nonlocal Fisher equation with the nonlocal diffusion term. Our second aim is to extend weakly nonlinear analysis to integro-difference equations. The multi-scale perturbation method has been used to obtain SL equations for the two proposed discrete-time integro-difference equations. Our computations show that transitions to instabilities are supercritical for symmetric nonlocal contact kernels. Thus one can observe that the amplitude of patterns near stability boundary is small. Hence using SL equations one can approximately compute the amplitudes of the patterns near stability boundary. The paper is organized as follows. We begin by detailing our models in the next section, and relate these models to the existing RDEs. In Section 3, we perform linear stability analysis of two models with periodic boundary conditions, and obtain conditions on the nonlocal interaction kernel for which solutions to our models exhibit pattern formation. In Section 4, we perform weakly nonlinear analysis of patterns and obtain amplitude equations. In Section 5, we summarize our findings and discuss further extensions of the methods developed in the paper.

Fisher (1937) proposed his famous equation to describe the spatial spread of an advantageous allele whose frequency is given below as a function of space and time vðx; tÞ: ð1Þ

where the Laplacian is used to model mobility of individuals and the reaction term is taken as logistic growth function. We consider a discrete-time model possessing many of the attributes of RDEs (Kot and Schaffer, 1986). We assume that growth and dispersal occurs during different life cycles. First consider an organism with nonoverlapping generations. The growth of such a population is governed by a nonlinear map unþ1 ¼ f ðun Þ

R

where K1 is the distance dependent dispersal kernel. This equation models dispersing individuals with non-overlapping generations. For logistic and Ricker's maps, (2) was studied by Kot and Schaffer (1986) and Kot (1992). Recently Fisher equation (1) with a nonlocal interaction term was investigated by many authors (Britton, 1989; Gourley, 2000; Genieys et al., 2006; Fuentes et al., 2003) in terms of pattern formation and traveling waves. The model is given as follows: vt ¼ Dv þ vð1  K 2  vÞ

ð3Þ

where the convolution term is as defined in (2) and K2 models nonlocal interactions. It was shown that the space homogenous solution of (3) becomes unstable if the Fourier transform of the kernel function K2 takes negative values. Thus, it was concluded that instability of space homogeneous solution is driven by the nonlocal interaction term. For derivation of the nonlocal reaction term in (3), one can consult with Genieys et al. (2006). Following the idea presented in Genieys et al. (2006), one can also include nonlocal competition term to density dependent growth functions governing the discrete time evolution of population densities. We will focus on the following two growth functions with nonlocal competition: The logistic growth function: f r ðu; K 2  uÞ ¼ ruð1  K 2  uÞ

ð4Þ

and The Ricker's growth function: f R ðu; K 2  uÞ ¼ ueRð1K 2 uÞ :

ð5Þ

Our aim is to investigate (2) with nonlocal interactions in the discrete time fashion. Hence we consider

2. The model

vt ¼ Dv þ vð1  vÞ

where f can be taken as logistic or Ricker's logistic growth functions. Above given map does not allow spatial movements of individuals. To consider these movements, we denote the density of infective individuals at spatial location x and time n by un(x). As mentioned above, in the sedentary step un(x) is mapped into f (un(x)). The second stage is spatial shuffling. This stage is called dispersal stage and modeled by an integral operator. To be able to model such movements we use a probability kernel K1(x,y) describing the dispersal of individuals from y. In particular, it determines the probability that an infective individual in an interval of length dy about y disperses to an interval of the same length about x. In a biological point of view, it is reasonable to assume that the dispersal weight depends on the relative distance i.e. K1(x, y) = K1(x  y). Hence we obtain a classical integrodifference equation modeling these two steps as follows: Z unþ1 ¼ ½K 1  f ðun ÞðxÞ ¼ K 1 ðx  yÞf ðun ðyÞÞ dy ð2Þ

unþ1 ¼ K 1  f j ðun ; K 2  un Þ;

ð6Þ

for j = l, R. Here the kernel function K1 models the spatial dispersal of individuals; and the kernel function K2 models the nonlocal interactions such as nonlocal competition for resources. The biological meaning of the nonlocal term (K2 * u) is that consumption of resources at a spatial location for mobile individuals does not only depend on the population density at that point but also on the weighted average of the population at the same point. One can consider paper wasps as an example of species having discrete non-overlapping generations with mobility. We aim to analyze the proposed model (6) with growth terms (4) or (5) with periodic boundary conditions. Our specific assumptions are summarized as follows:

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 We assume that the kernel functions are compactly supported.  We adopt the work of Neubert et al. (1995) by considering symmetric and positive kernel functions.  We assume that there is no mortality during dispersal or interaction similar to RDEs (Gourley, 2000; Genieys et al., 2006; Aydogmus, 2015) with nonlocal interaction term. In other words, dispersal and interaction kernels are probability density functions on the spatial space. More specifically, our assumptions regarding dispersal and interaction kernels are given as follows: For all x 2 R and j = 1, 2,

compact. In addition our assumptions guarantee that the operator is self-adjoint and hence, there exists at least one non-zero eigenvalue. Thus, its spectrum consists of at most the point zero and countable number of non-zero eigenvalues. Following the method used in Kot and Schaffer (1986), the solution to the linear problem jt(x) can be assumed to have the following form

jt ðxÞ ¼ lt j~ðxÞ:

ð8Þ

By using the expression above, we can obtain the explicit form of solution to (7) by considering time-independent equation corresponding to the linear problem (7) as follows:

lj~ ¼ K 1  ½j~  cK 2  j~; for x 2 ½0; L:

 Kj(x)  0 is compactly supported and piecewise continuous,  Kj(x) = KJ (x), R  R K j ðxÞ dx ¼ 1. Note that, in the absence of space, the unique interior is globally stable equilibrium of the logistic equation ul ¼ r1 r when 1 < r < 3; and the unique interior equilibrium of the Ricker's equation uR ¼ 1 is globally stable when 0 < R < 2. These constant solutions are also spatially homogeneous solutions of (6) with growth terms (4) or (5), respectively. In the following section, we investigate the stability conditions of the constant solutions ul and uR for the respective model. 3. Linear analysis

n

where cn can be determined by a Fourier expansion of the initial ~, and l condition in terms of the time-independent solution j denotes the eigenvalue for the temporal growth. By plugging the solution of the form (10) into Eq. (7), one obtains

Eq. (6) with growth terms (4) or (5) admits constant (spatially homogeneous) solutions ul and uR which are globally stable for the corresponding model without space when r 2 (1, 3) and R 2 (0, 2), respectively. To investigate the effect of nonlocal interactions on stability of the space homogenous solutions to (6) with growth terms (4) and (5), we restrict ourselves to the parameter spaces r 2 (1, 3) and R 2 (0, 2), respectively. Linearization of these equations can be obtained by using the small perturbation of the equilibrium points as below:

ð11Þ

for n 2 Z. Here K^1 ðkn Þ and K^2 ðkn Þ are Fourier transforms of the kernel functions K1 and K2 evaluated at kn. Thus stability condition for the equilibria ui for i = l, R is given as |l(c, kn)| < 1 for all kn, n 2 Z. This implies that instability of ^ 1 ðkn Þjj1  cK ^ 2 ðkn Þj  1 for some equilibrium u or u requires jK l

3.1. Linear stability analysis

ð9Þ

The eigenfunctions of this time-independent equation are given by exp(iknx) satisfying the boundary conditions specified above with kn ¼ 2pL n for n 2 Z. Hence one can write the solution of (7) in terms of eigenfunctions as follows: X t jt ðxÞ ¼ cn l expðikn xÞ ð10Þ

lðc; kn Þ ¼ K^1 ðkn Þð1  cK^2 ðkn ÞÞ

In this section, we perform linear stability analysis of Eq. (6) with growth terms (4) and (5) to obtain conditions that destabilize the constant fixed points (i.e., the space homogenous solution) of these two models. In addition, we present some computational results.

3

R

^ 1 j  1, the dispersal kernel K1 stabilizes the n 2 Z. Since jK ^ j for j = 1, 2 are nonlinear functions, it is not equilibrium. Since K possible to obtain exact conditions to observe pattern formation without knowing the exact form of the dispersal kernel. For this ^ 2 ðkn Þj  1 for some n 2 Z to be able reason, we require that j1  cK to gain some information regarding the property of the interaction kernel. Note that if the Fourier transform of the interaction kernel ^ 2 ðkn Þ > 1. Hence a takes negative values for some n 2 N then 1  cK

Plugging this into equations gives the following first order relation

necessary condition to destabilize the space homogenous equilib^ 2 ðkn Þ < 0 for ria of Eq. (6) with growth terms (4) and (5) is that K some n 2 N. Let c0 2 (0, 2) be the smallest number such that there exists a wavenumber kn0 satisfying the following maximization problem:

jtþ1 ¼ K 1  ½jt  cK 2  jt 

maxn2Z jlðc; kn Þj ¼ lðc0 ; kn0 Þj ¼ 1:

ut ¼ uj þ ejt

for j ¼ r; R:

ð7Þ

where c 2 (0, 2) is equal to r  1 if logistic growth (4) is used and R if Ricker's growth (6) is used. Following the approach in Neubert et al. (1995), we perform a linear stability analysis of Fourier transformed linear relation corresponding to (7). In Neubert et al. (1995), linear stability analysis is performed in an infinite domain and hence no boundary conditions were used. Our aim is to analyze the problem with periodic boundary conditions on a finite interval. Thus, we consider the linear problem (7) with wrap around boundary conditions in a box D = [0, L] i.e.

jt ðxÞ ¼ jt ðx þ LÞ; for all t 2 N and x 2 D: For finite integration limits due to the periodic boundary conditions and continuous kernels, linear operator (7) (Frechet derivative of corresponding nonlinear Hammershtein operator) is

ð12Þ

Then we call c0 as the critical value of the parameter c, and kn0 as the related critical wavenumber. We summarize our findings in the following proposition. Proposition 1. Suppose that K 2 ðkn0 Þ < 0 and there exists a constant c0 2 (0, 2) satisfying equality (12). Then jlðc; kn0 Þj  1 and solutions to (4) and (6) exhibit pattern formation for any c  c0. Notice that jlðc0 ; kn0 Þj ¼ 1 implies that lðc0 ; kn0 Þ ¼ 1. As done by Neubert et al. (1995), one can classify the possible bifurcations as follows:  If lðc0 ; kn0 Þ ¼ 1, then lðc; kn0 Þ > 1 for c > c0 and uniform steady state loses its stability to a spatially structured steady state This is referred as a plus one bifurcation.

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 If lðc0 ; kn0 Þ ¼ 1, then lðc; kn0 Þ < 1 for c > c0 so that the stability is lost to a spatially structured time periodic solution of period two. This is referred as a minus one bifurcation. Eq. (11) implies that both of these bifurcations are possible depending on the sign of K 1 ðkn0 Þ. However, our numerical work for different symmetric kernel functions studied in Segal et al. (2013) and Neubert et al. (1995) suggests that only plus one type bifurcation occurs. 3.2. Computational results It has been shown by Genieys et al. (2006) that solutions to nonlocal Fisher equation (3) exhibit pattern formation provided that K 2 ðkn0 Þ < 0. Hence, symmetric uniform kernels have been used extensively in the literature (see for example Fuentes et al., 2003, 2004; Genieys et al., 2006). One can consider the family of symmetric uniform kernel functions as follows: ! ( 1 if xj  d K d ðxÞ ¼ 2d ð13Þ 0 if jxj > d; where the parameter d is called the range of nonlocality of the kernel function K (Segal et al., 2013). Gaussian or Laplacian kernel families are other popular kernel functions widely used in literature (Neubert et al., 1995; Kot and Schaffer, 1986; Kot, 1992; Tanzy et al., 2013). Contrary to these two families, the Fourier transform of uniform kernels K d takes negative values even if the kernel is symmetric. Thus uniform kernel family K d satisfies the necessary conditions to observe pattern formation as indicated in Proposition 1. Before proceeding the numerical results, we consider the following extreme cases for dj ! 0 or L for j = 1, 2. Let's denote the RL spatial average of the population densities by aðuÞ ¼ 1L 0 uðyÞ dy.  Suppose that d2 ! 0 in Eq. (6) for growth functions defined by (4) and (5) then the kernel K2 approaches to dirac delta function and the nonlocal interaction term becomes localized i.e. K2 * u ! u. In this case (6) is equivalent to (2).

In addition, if d1 ! 0 then we have the logistic or Ricker's type growth at any spatial location x without any spatial structure. Hence ut ! u* as t ! 1.  If, on the other hand, d1 ! L then ut+1 = a(fj(ut)) for j = l, R. Note that ut is a constant function in spatial variable x and hence ut = a (ut) for t  1. This implies that we have a(ut+1) = fj(a(ut)) for t  0. The average follows a logistic or Ricker's type growth implying ut = a(ut) ! u* as t ! 1.

(ut)). The average follows a logistic or Ricker's growth implying that ut = a(ut) ! u* as t ! 1. As mentioned above, solutions to these two integro-difference equations exhibit pattern formation if there is a critical value c0 2 (0, 2). Here, we solve the optimization problem (12) with different starting points for the uniform kernel family (13) to find the critical parameter values c0. In particular, we restrict ourselves to the following region of parameters:  K j ¼ K dj with 0.2  d1 1 and 0 < d2  4. For the parameter space of 0.2  d1 1 and 0 < d2  4 in both models (4) and (6), white region in Fig. 1 represents the values of parameters when one can observe pattern formation i.e, there exist a critical parameter value c0 2 (0, 2) and spatially homogeneous stable solutions ui for i = l, R become unstable for any parameter c > c0. On the other hand, one cannot find such a critical parameter c0 2 (0, 2) for the parameter region colored in black in Fig. 1. In Fig. 1, the instability region (white region) is obtained numerically in terms of ranges of nonlocalities of dispersal and nonlocal competition kernels d1 and d2. For any fixed value of range of nonlocality for interaction kernel d2 one can easily observe that decreasing the dispersal range d1 results in formation of spatial patterns. Note that nonlocal growth terms (4) and (5) depends both on local population density and nonlocal starvation term with carrying capacity 1. Since starvation term depends the weighted average of population density, local population at some spatial position x may increase depending on the available resources in the neighborhood of x (short range activation) even if the local population density is larger than the carrying capacity. It is also quite general observation that increasing the dispersal range of the population stabilizes the space homogenous equilibrium. Similar results for other ecological models has been obtained by Doebeli (1995), Hastings (1993) and Gyllenberg et al. (1993). On the other hand fix a value of dispersal range d1. The parameter d2 measures the range of nonlocality for nonlocal competition term. As d2 decreases the spatial average term [K2 * u] (x) becomes more concentrated about the spatial location x and approaches to delta function, whereas as it increases the density of the population further away becomes more important for the evolution of the population density ut. Therefore, aggregation

 Suppose that d2 ! L in Eq. (6) for growth functions defined by (4) and (5) then the convolution K2 * ut is equal to a(ut). In this case (6) is equivalent to following integro-difference equation for j = l, R: utþ1 ¼ gj ð1  aðut ÞÞK 1  ut ; where gl(u) = r(1  a(u)) and gR(u) = Re1a(u).  Now consider the case d1 ! 0 then we have ut+1 = gj(1  a(ut))ut for j = l, R. By taking the spatial average of both sides one can see that a(ut) obeys logistic or Ricker's type growth. Hence a(ut) ! u* as t ! 1. Note that the spatial average approaches to spatially uniform equilibrium, but it does not mean that the solution is spatially homogenous in general.  Lastly, consider the case d1 ! L then ut+1 = gj(1  a(ut))a(ut) for j = l, R. Note that ut is a constant function in spatial variable x and hence ut = a(ut) for t  1. This implies that we have a(ut+1) = fj(a

Fig. 1. Parameter region P where solutions to (4) and (6) exhibit pattern formation for some values of c > c0 for some c0 2 (0, 2). White region represents the values of parameters when pattern formation occurs while black region represents the values of parameters when one cannot find any c0 2 (0, 2).

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5

occurs if the range of nonlocality for the interaction kernel is large enough i.e. the inhibitive effect of crowding is sufficiently long range. In our numerical simulations, we consider (6) for growth terms (4) and (5) with wrap around boundary conditions. Both convolution terms K1 * u and K2 * u are approximated by fast Fourier transforms. We used the discretization of the space variable 0  x  L with a mesh interval of Dx = 0.0001 and L = 10. Near the stability boundaries, we consider the following set of parameters: both panels (a) and (b) in Fig. 2 are obtained for the parameters d1 = 0.25 and d2 = 1 with initial condition u0 ¼ 0:001cosðkn0 xÞ and c = 1.0613. For above specified kernel functions, the critical value of the parameter c0 is given by 1.0581.

We are interested in the stability of the spatially constant steady  states ul ¼ r1 r and uR ¼ 1 of (6) with growth terms (4) and (5) for 1 < r < 3 and 0 < R < 2. Note that the spatially homogeneous steady state ul of Eq. (4) does depend on the parameter c = r  1. Therefore a perturbation of the parameter c (or r) will induce a perturbation of steady state ul . Denote the steady state corresponding to the critical value r0 by u0. Then one can easily observe that the steady state corresponding the parameter r = r0 + e2n can be written as

4. Weakly nonlinear analysis of patterns

Em ¼ expðimkn0 xÞ:

The linear stability analysis is only valid for small time and infinitesimal perturbations. To analyze the behavior of the growth of unstable modes we should account the nonlinear terms. Let c0 be the critical value of parameter c for which lðc0 ; kn0 Þ ¼ 1. Solution of (6) with growth terms (4) and (5) is given as follows:

Using these linear waves we construct a solution to (6) with growth term (4) as follows:

u / l expðikn0 xÞ þ c:c: t

ð14Þ

where c.c. stands for complex conjugate and i denotes the complex unit number. We perform a perturbation analysis near the critical value c0: c ¼ c0 þ e2 n;

0 < e  1; n ¼ 1

ð15Þ

Clearly the eigenvalue lðc; kn0 Þ is real. Using the above perturbaand mean value theorem, one gets tion lðc; kn0 Þ ¼ 1 þ ne2 l0 ðc ; kn0 Þ. Substituting this extension into Eq. (14) gives us   expðikn0 x þ tlogðlðc; kn0 ÞÞÞ ¼ exp ikn0 x þ tlogð1 þ ne2 l0 Þ   2 0 ¼ exp iðkn0 xÞ þ tne l Aðe2 tÞexpðiðkn0 xÞÞ Hence the amplitude A is a function of the slow time t : = e2t. We consider the fast and slow time scales t and t together in our analysis. Following Holmes (2012), we incorporate the slow time scale t into the problem by assuming the solution has an expansion of the form ut ðxÞ ¼ u0 ðt; t ; xÞ þ eru1 ðt; t ; xÞ þ    : The functions uj for j = 0, 1, 2 will be determined in the following lines.

ul ¼ u0 þ e2

n r20

þ Oðe4 Þ

. where u0 ¼ r0r1 0 Now consider the family of linear waves:

uðt; t ; xÞ ¼ u0 þ e2

n r20

þ eu1 ðt; t ; xÞ þ e2 u2 ðt; t ; xÞ

ð16Þ

where u1 ¼ Aðt ÞE1 þ c:c:

and

u2 ¼ A0 ðt Þ þ ðA2 ðt ÞE2 þ c:c:Þ:

Without loss of generality, we assume that A0 is real. Hence we do not consider the complex conjugate of the term A0 in our analysis. In a similar manner one can also construct a solution to (6) with growth term (5) as follows: uðt; t ; xÞ ¼ 1 þ eu1 ðt; t ; xÞ þ e2 u2 ðt; t ; xÞ

ð17Þ

where u1 and u2 are as given above. By performing a perturbation analysis, we obtain the following Stuart–Landau type equation as follows: At ¼ nj A þ j A3

ð18Þ

for j = r, R. This equation informs us about transitions to instability and amplitudes of patterns. For the derivation of this equation see Appendix A. The coefficients Fl and Cl are given in the appendix by Eqs. (A.6) and (A.7). Similarly FR and CR are given by the formulas (A.8) and (A.9). Since coefficients _FPhii and C Psii for r, R of Stuart–Landau equation (18) are real, the amplitude A can be taken as a real function provided that the initial amplitude A(0) is real. Steady state solutions of Eq. (18) are given by A0 = 0 and A1i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nFj =Cj for J = r, R. The linear stability analysis of (18) suggests

Fig. 2. Panel (a) shows stationary wave type patterns arising from Eq. (4) with parameters d1 = 0.25, d2 = 1 and c = 1.0613. Panel (b) illustrates stationary wave type solutions to Eq. (6) for the same parameters.

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that the solution 0 is stable whenever n Fi < 0. We can observe that Fi > 0 from equality (A.2). Thus, we can conclude that the steady state solution A0 (or ui for i = l, R) is unstable for n = 1 and it is stable for n =1. This result is in consistence with the results of the linear analysis in Section 3. Moreover, we need Af1; jg to be real i.e. n C Psii  0 for j = r, R. Thus the sign of C Psii determines the direction of deviation from bifurcation parameter. For the parameter set P, we observed that C Psii < 0 for j = r, R. Hence we only get supercritical bifurcations for amplitudes of patterns arising from both models (4) and (6). The summary of our findings can be given as follows:  Our computations for parameter set P indicates that C Psii for J = r, R take negative values. Hence m = 1 and (18) describes a supercritical bifurcation with stable branch eA1 to the right of c0 for both models (4) and (6). To be able to verify these results numerically, define the amplitude of solutions to Eq. (4) and (6) by Amp :¼ maxx2I fuðx; t Þg  minx2I fuðx; t Þg

ð19Þ

*

for sufficiently large t . We study quantity Amp defined above and compare it to the numerical observations. By substituting the perturbation extension (16) and (17) with the first two terms i.e. u = 1 + eu1 and u0 + eu1 to (19). Thus, we can obtain Amp ¼ 4eA1 :

ð20Þ

We verify our analytical results obtained in this section by numerical simulations. Fig. 3(a) and (b) shows the stable branch (black curve) obtained by using the analytical results and stable pattern amplitudes (black dots) for small perturbations of critical parameter c0. Fig. 3(a) shows the amplitude levels of patterns arising from (4) for parameters d1 = 0.25 and d2 = 1. Fig. 3(b) shows the amplitude levels of patterns arising from (6) with the same parameters. The initial condition is again taken as u0 ¼ 0:001cosðkn0 xÞ. As seen from the figure, nonlinear prediction and the amplitude A obtained from numerical simulations of (4) stay pretty close for small values of e. As a result of weakly nonlinear analysis we only find continuous transitions to instability for symmetric dispersal and competition kernels. Hence, one can only observe spatial heterogeneity if the bifurcation parameter (or growth rate) c is larger than its critical value c0. Note that in the case of discontinuous transitions to instability one can also observe patterns when c  c0 (Topaz et al., 2006; Eftimie et al., 2009; Aydogmus, 2015). Here one can easily pffiffiffiffiffiffiffiffiffiffiffiffiffi observe that the amplitude of the patterns is of order e ¼ c  c0. In other words, as the growth rate increases, the amplitude of the

patterns increases. Hence increasing the growth rate supports aggregation. 5. Discussion In this study, we considered two simple intraspecific competition models with non-overlapping generations. In our models both individuals’ movement and interaction are governed by probability kernel functions modeling distance-dependent interactions and movement patterns. Both of these nonlocal terms were assumed to depend on a parameter that is the range of nonlocality. We investigated the necessary conditions to destabilize space homogeneous solutions uj for j = r, R in terms of the Fourier transforms of interaction and dispersal kernels. We observed the emergence of patterns using linear stability and computational analyses. Our ecological reasoning to add nonlocal competition term to logistic and Ricker's type equations is that the growth term for mobile individuals should depend on the spatially weighted average of resources available as mentioned by Britton (1989). Hence the carrying capacity of the spatial habitat is not only a function of the local population density but the weighted average of the population densities in a neighborhood of the spatial location at which point the animal disperses. Increasing the range of nonlocality, as shown by Fig. 1, favors aggregation. Such a long range inhibitive effect of crowding supports local growth (short range activation) of the species. Hence the population density may continue to increase even if the current density is above the carrying capacity resulting in spatially heterogenous patterns. On the other hand, we observed that spatial heterogeneity vanishes as dispersal range increases. We further analyzed these models near stability boundaries. As a result of weakly nonlinear analysis we obtained Stuart–Landau equations informing us about the amplitude of patterns arising from our model. Using the multi-scales analysis and computations, we show that transition to instability is supercritical for parameter set P. We also numerically verify the results of nonlinear analysis. Near critical bifurcation parameter, the result of weakly nonlinear analysis implies that as the growth rate c increases the amplitude pffiffiffiffiffiffiffiffiffiffiffiffiffi levels of patterns increases and is of order c  c0. Hence we conclude that the larger growth rate favors aggregation. When we consider the classical integro-difference equation (2), it is obvious that there is no advantage to individuals in the population in grouping together. Hence the classical model (2) contains no mechanism for aggregation. The results obtained in this paper imply that spatially nonlocal interactions can be considered as a mechanism for aggregation. The linear and nonlinear analyses presented in this paper enables us to determine the effect of our model parameters c and dj for j = 1, 2 on aggregation. Increasing the dispersal range d1 has a negative effect

Fig. 3. Numerical verification of analytical results. Panel (a) shows amplitude levels of stationary waves with parameters d1 = 0.25 and d2 = 1 for the logistic model (4). Panel (b) shows the amplitude levels of patterns with the same parameters for the Ricker's model (6).

Please cite this article in press as: O. Aydogmus, et al., Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities, Ecol. Complex. (2017), http://dx.doi.org/10.1016/j.ecocom.2017.04.001

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Fig. 4. Traveling wave type patterns arising from Eq. (6) for the interaction kernel K d;a with d = 0.45 and a = 0.05. Panel (a) is obtained using logistic growth function (4) with parameters d1 = 0.1, and c = 1.3. Panel (b) illustrates the obtained patterns using Ricker's type growth function (5) for the same parameters.

on aggregation while increasing the range of nonlocal competition d2 supports spatially heterogenous solutions. The result of nonlinear analysis, on the other hand, implies that increasing the growth rate results in more denser groups of individuals. Finally there are a few interesting issues that should be further investigated. One further study is extending the analysis to non self-adjoint operators. To be able to present numerical results regarding such operators, we consider the following uniform family of kernel functions: ! ( 1 ð21Þ K d;a ðxÞ ¼ 2d if jx  aj  d 0 if jx  aj > d: Using the same computational setting presented in Section 3.2, we obtained the traveling wave type patterns arising from the model (6) with both logistic and Ricker's type growth. To be able to obtain such patterns we used K 1 ¼ K 0:1;0 and K 2 ¼ K 0:05;0:45 (Fig. 4). Here a possible further study might be considering the problem of constructing the linear waves for non self-adjoint operators and obtaining a SL equation for traveling wave type patterns. In this case, the corresponding SL equation will have complex coefficients. In addition, weakly nonlinear analysis for integro-difference equations can be extended to systems of equations and amplitude levels of dispersal driven instabilities can also be studied via this method. Lastly, it is also possible to extend the weakly nonlinear analysis to integro-difference equations defined on two space dimensions to be able to analyze the properties of patterns.

Acknowledgements The authors thank two referees for their valuable suggestions which led to a significant improvement over an earlier version of the manuscript. This research of YK is partially supported by NSF-DMS (Award Number 1313312), NSF-IOS/DMS (Award Number 1558127) and The James S. McDonnell Foundation 21st Century Science Initiative in Studying Complex Systems Scholar Award (UHC Scholar Award 220020472).

Appendix A. Derivation of SL equations A.1 Derivation of SL equation for logistic model Before performing the perturbation analysis, we define the following notations for the kernel functions. ^ 1n K^1 ðnkn0 Þ ¼ K

and

^ 2 ðnkn Þ ¼ K ^ 2n : K 0

^ iðnÞ ¼ K ^ in for any symmetric kernel In addition, note that K function Kj, j = 1, 2. Here we also would like to note that the ^ jn En is used implicitly for any n 2 Z and j = 1, 2. identity K j  En ¼ K Then we consider ansatz (16) and expand the term un+1 following Holmes (2012) as follows: unþ1

¼

uðt þ 1; t þ e2 ; xÞ

¼

u0 þ e2

n

þ eu1 ðt þ 1; t ; xÞ c20 2 þ e u2 ðt þ 1; t ; xÞ þ e3 @t u1 ðt þ 1; t ; xÞ þ Oðe4 Þ:

ðA:1Þ

By plugging the ansatz (16) in Eq. (4) and using the expansion (A.1), one can obtain ^ 21 Þ  AÞE1 þ c:c: ¼ 0 ^ 11 ðA þ ð1  r0 ÞAK ðK

ðA:2Þ

at O(e) level. Note that at this level, one can recover the linear relationship (11) as in the case of continuous time systems (Aydogmus, 2015; Eftimie et al., 2009; Topaz et al., 2006). At the level O(e2), we obtain the following relationship:   ^ 21 A2 þ ð1  r0 ÞA2 K ^ 22 Þ  A2 E2 þ c:c: ^ 12 ðA2  r0 K K   ^ 21  A0 ¼ 0 þ 2A0  r0 A0  2r0 jAj2 K where |A| denotes the norm of the complex function A. By requiring the coefficients of the eigenfunctions E0 and E 2 are 0, we obtain: A0 ¼

2r0 ^ K 21 jAj2 1  r0

ðA:3Þ

Please cite this article in press as: O. Aydogmus, et al., Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities, Ecol. Complex. (2017), http://dx.doi.org/10.1016/j.ecocom.2017.04.001

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References A2 ¼

^ 21 K ^ 12 r0 K A2 : ^ 22 K ^ ^ 12  1 K 12 þ ð1  r0 ÞK

ðA:4Þ

At O(e3)E1 level, one can easily obtain the following equality:   ^ 21  r0 ð1 þ K ^ 21 ÞA0 A Þ r0 ðK ^ 22 þ K ^ 21 ÞA A2  At ¼ 0: ^ 11 nAK K ðA:5Þ Thus, using equalities (A.3) and (A.4) in (A.5), one can get Stuart–Landau type equation (18) associated to Eq. (4) with coefficients:

Fl ¼ K^ 11 K^ 21

ðA:6Þ

Cl ¼ r20 K^ 11 K^ 21 C l where C l ¼



^ 22 þK ^ 21 Þ ^ 12 ðK K ^ 22 K ^ 12 þ1 ^ 12 K ðr 0 1ÞK

ðA:7Þ þ



^ 21 Þ 2ð1þK r0 1

.

A.2 Derivation of SL equation for Ricker's model As done in Appendix A we use the expansion (A.1) and ansatz (17) to obtain coefficients of SL equation for Ricker's model (6). At O (e) level one recovers the linear equation again. Equalities from O(e2) we get the following equalities: A2 ¼

^ 12 K ^ 21 ð2  R0 K ^ 21 Þ 2 R0 K A ^ ^ ^ 2ðK 12  R0 K 12 K 22  1Þ

and ^ 21 ðR0 K ^ 21  2ÞAAc : A0 ¼ K

At

¼

^ 21 ÞAA0 ^ 21 A  R0 ð1  R0 K ^ 21 þ K nK ^ 22  K ^ 21 ÞA2 A ^ 21 K ^ 22  K þR0 ðR0 K 3 ^ 2 ð3  R0 K ^ 21 Þ A þR20 K 21 2

Using above equalities one can obtain the parameters for SL equation as follows:

FR ¼ K^ 11 K^ 21

ðA:8Þ

CR ¼ R0 K^ 11 K^ 21 C R

ðA:9Þ

where CR

¼

^ 21 þ K ^ 21 Þ ^ 21 Þð2  R0 K ð1  R0 K ^ ^ ^ 22  K ^ 21 Þ R0 K 12 ð2  R0 K 21 Þ ^ 21 K ^ 22  K þðR0 K ^ 12  R0 K ^ 12 K ^ 22  1Þ 2ðK ^ ^ 21 3  R0 K 21 : þR0 K 2

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Please cite this article in press as: O. Aydogmus, et al., Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities, Ecol. Complex. (2017), http://dx.doi.org/10.1016/j.ecocom.2017.04.001