- Email: [email protected]
Factors promoting or inhibiting Turing instability in spatially extended prey–predator systems
Ecological Modelling 222 (2011) 3449–3452
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
Factors promoting or inhibiting Turing instability in spatially extended prey–predator systems Stefano Fasani a , Sergio Rinaldi a,b,∗ a b
DEI, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy Evolution and Ecology Program, International Institute for Applied Systems Analysis, 2361 Laxenburg, Austria
a r t i c l e
i n f o
Article history: Received 27 January 2011 Received in revised form 29 June 2011 Accepted 3 July 2011 Available online 28 July 2011 Keywords: Prey–predator model Spatial pattern Turing instability Diffusive instability Dispersal
a b s t r a c t The emergence of inhomogeneities in the distributions of the abundances of spatially extended prey–predator systems is investigated. The method of analysis, based on the notion of diffusive (Turing) instability, is systematically applied to nine different models obtained by introducing an extra-factor into the standard Rosenzweig–MacArthur prey–predator model. The analysis confirms that the standard model is critical in the context of Turing instability, and that the introduction of any small amount of the extra-factor can easily promote or inhibit the emergence of spatial patterns. © 2011 Elsevier B.V. All rights reserved.
1. Introduction As it is well known, plant and animal populations are not uniformly distributed in the environment but are packed in patches that remain fixed or move over time (spatio-temporal patterns). The most obvious reasons for patchiness are the occurrence of local perturbations (e.g. the sudden discharge of a chemical species at a specific point) and the presence of permanent spatial inhomogeneities in the environment (e.g. better exposition to sun of one of the river banks). A much more subtle reason for patchiness in ecology has been pointed out long ago by Segel and co-workers (Segel and Jackson, 1972; Levin and Segel, 1976) who used the idea of diffusive instability and the results obtained by Turing (1952) to show that simple prey–predator interactions can be responsible of patchiness even in perfectly homogeneous environments. However, the prey–predator models used by Segel are rather special and not very convincing, in particular in the context of plankton dynamics (Levin and Segel, 1976). Most likely, the reason for this is that more standard models, like the Rosenzweig–MacArthur model (Rosenzweig and MacArthur, 1963), are not appropriate for proving the existence of diffusive instability. In fact, as recognized later in
∗ Corresponding author at: DEI, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy. Tel.: +39 02 2399 3563; fax: +39 02 2399 3412. E-mail address: [email protected] (S. Rinaldi). 0304-3800/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2011.07.002
Alonso et al. (2002), the Rosenzweig–MacArthur model is “critical” in the context of Turing’s theory. More precisely, the two biological mechanisms taken into account in the Rosenzweig–MacArthur model, namely logistic growth of the prey and saturating functional response of the predator, do not imply the emergence of diffusive instability, unless, as shown in this paper, a small amount (actually any small amount) of a suitable extra-mechanism is also included in the model. Since the Rosenzweig–MacArthur model is one of the most popular prey–predator models in ecology, the identification of the extra-mechanisms that must be added to it to obtain a model supporting pattern formation, is an interesting problem. After recalling the basic Turing results and the interpretation given to them in Satnoianu et al. (2000) and in Della Rossa et al. (2011), we show how the problem can be solved with almost no effort. The result of the analysis is that all elementary mechanisms not taken into account in the Rosenzweig–MacArthur model can be partitioned in three classes:
(i) Mechanisms, like predator cooperation or exploitation, promoting diffusive instability if predator disperse sufficiently less than prey. (ii) Mechanisms, like predator interference, intraspecific competition and stocking, always inhibiting diffusive instability. (iii) Mechanisms, like prey harvesting or stocking, with absolutely no influence on diffusive instability.
3450
S. Fasani, S. Rinaldi / Ecological Modelling 222 (2011) 3449–3452
Table 1 Nine independent factors p not taken into account in the Rosenzweig–MacArthur model (4) (first column) and the predator per-capita growth rate g(x, y, p) that should be used in model (2) to take these factors into account (second column). The sign of ∂g/∂y (third column) allows one to identify if the factor p promotes (+) or inhibits (−) diffusive instability. Factor p
Predator per-capita growth rate g(x,y,p)
Sign of
1. Predator harvesting 2. Generalist superpredator
ax e 1+ahx − d − p 1+a1 h y ax 1 e 1+ahx − d − p 1+a h y+a h W
+ +
3. Predator cooperation
a(1+py)x e 1+a(1+py)hx −d
+
4. Predator interference 5. Predator intraspecific competition 6. Predator stocking
ax e 1+ahx+py −d ax e 1+ahx − d − py ax e 1+ahx − d + py
− − −
7. Prey harvesting 8. Prey stocking
ax e 1+ahx −d ax e 1+ahx −d
0 0
a(1−px)x e 1+a(1−px)hx −d
9. Prey cooperation
2. Conditions for diffusive (Turing) instability Spatially extended prey–predator systems can be described, under the assumption of diffusive dispersal, by a PDE of the form ∂x ∂t ∂y ∂t
=
f (x, y, p)x + dx ∇ 2 x
=
g(x, y, p)y + dy ∇ 2 y
(1)
where x and y are prey and predator abundances depending upon time and space in a given domain of R2 , f and g are per-capita growth rates depending upon demographic parameters p, and dx and dy are the non-negative dispersal coefficients of the two populations. Typically, in order to have a well posed problem, zero-flux or periodic boundary conditions are also imposed. ¯ of Eq. (1) A positive, homogeneous and stationary solution (¯x, y) (characterized by f = g = 0) can be stable in absence of diffusion (i.e. for dx = dy = 0) but unstable for suitable pairs (dx ,dy ). In other words, ¯ can be stable in a lumped prey–predator model an equilibrium (¯x, y) dx dt dy dt
=
f (x, y, p)x
=
g(x, y, p)y
(2)
but unstable in its spatially extended version Eq. (1). This somehow counterintuitive phenomenon, first investigated in a celebrated paper of Turing (1952), is known as diffusion-induced instability, but is also called Turing instability. The original Turing analysis considers the effect of small pertur¯ in the Fourier expansion of the solution bations imposed on (¯x, y) of Eq. (1) and the key result (Satnoianu et al., 2000) is that diffusion induced instability is equivalent to the instability of the matrix C = A(p) − D
(3)
¯ i.e. where A(p) is the Jacobian of Eq. (2) at the equilibrium (¯x, y),
⎡ ∂f x¯ A(p) = ⎣ ∂x y¯
∂g ∂x
∂f ∂y ∂g y¯ ∂y x¯
⎤ ⎦
and D is the diagonal matrix with dx and dy as diagonal elements. If dx = dy , i.e. if D is proportional to the identity matrix, the spectrum of C in Eq. (3) is simply the spectrum of A shifted to the left, so that C cannot be unstable if A is stable. This is why the dispersion coefficients must be unbalanced in order to have Turing instability. As shown in (Della Rossa et al., 2011; Satnoianu et al., 2000), the ¯ is stable in Eq. problem of finding triplets (p,dx ,dy ) for which (¯x, y) (2) but unstable in Eq. (1) can be solved in two steps, namely: (i) find values of p for which one diagonal element of the Jacobian matrix A is positive. Notice that if aii is positive (i.e. if i is the
∂g ∂y
0
/ i, is negative, since A is stable so-called activator) then ajj , j = in Eq. (2) so that tr(A) must be negative. (ii) determine the dispersal coefficients (dx , dy ) realizing Turing instability by imposing that C is unstable. As shown in Satnoianu et al. (2000), this is always possible if the activator disperses sufficiently less than the other population (inhibitor). This decomposition, which has never been systematically exploited in the literature (see Segel and Jackson (1972), Levin and Segel (1976), Chakraborty et al. (1996), Alonso et al. (2002), Baurmann et al. (2007), Wang et al. (2007), Banerjee (2010), Banerjee and Petrovskii (2010), Sun et al. (2009), Sun et al. (2008), Wang et al. (2010), Zhang et al. (2009)), greatly simplifies the analysis, because the two steps (i) and (ii) can be easily accomplished. Moreover, step (ii) is often not needed in ecology, where the problem of major concern is the identification of the factors promoting or inhibiting diffusion-induced instability. 3. Criticality of the Rosenzweig–MacArthur model One of the most standard prey–predator models (Rosenzweig and MacArthur, 1963) is composed of a logistic prey and a type II predator and is therefore described by
x ay dx = r 1− − x K dt 1 + ahx dy ax = e −d y dt 1 + ahx
(4)
where r and K are prey net growth rate and carrying capacity and a, h, e and d are predator attack rate, handling time, efficiency and death rate, respectively. Model (4) in the form of Eq. (2) and ¯ for suitable values of its has a positive stable equilibrium (¯x, y) parameters. Since the per-capita growth rate g does not depend on predator density, the second diagonal element of the Jacobian A in Eq. (3) is zero, i.e. a22 = 0 for all parameter values, while a11 < 0 since tr(A) < 0 at a stable equilibrium. This means that Turing condition (a11 a22 < 0) is not satisfied in model (4), since in that model a11 a22 = 0. In other words, the Rosenzweig–MacArthur model is critical in the context of Turing instability. 4. Small factors can trigger Turing instability The addition to model (4) of any small amount p of an extra ¯ as well demographic factor slightly modifies the equilibrium (¯x, y) as the elements a11 and a22 of the matrix A. Hence, by continuity, the ¯ remains positive and stable and a11 remains negequilibrium (¯x, y) ative. By contrast, depending upon the p-factor, a22 , which is zero for p = 0, can be positive or negative (or equal to zero) for small values of p. In the first case (a22 (p) > 0 for small values of p > 0) the p-factor promotes Turing instability,i.e. pattern formation, if
S. Fasani, S. Rinaldi / Ecological Modelling 222 (2011) 3449–3452
3451
Fig. 1. Stationary spatial patterns of predator density obtained through simulation of model (1) on a squared spatial grid with periodic boundary conditions. First row: factor 1 in Table 1. Second row: factor 2 in Table 1. Third row: factor 3 in Table 1. Common reference parameter values: r = K = e = 1, h = 3, d = 0.2. Other parameter values: first row, p = 0.09, dx = 0.005, a = h = 1, k = 1000, a = 0.95,1,1.05 from left to right. Second row: p = 0.1, dx = 0.005, a = a = a = 1, h = h = 3, W = 0.05, k = 500,1000,1500 from left to right. Third row: k = 10,000, dx = 0.01, a = 0.9, p = 0.20,0.22,0.24 from left to right.
predator disperse sufficiently less than prey. The conclusion is that the factors promoting Turing instability can be simply identified by looking at the sign of ∂g/∂y at equilibrium. We now consider nine independent factors that are not taken into account in model (4) but can be easily added to it by modifying the predator per capita growth rate. Table 1 reports the nine factors, the functions g(x,y,p) and the signs of ∂g/∂y. Of course g(x, y, 0) is the Rosenzweig–MacArthur predator growth rate in all cases. For the first three factors, ∂g/∂y is positive for all parameter values, so that one can conclude that even the existence of a small amount of predator harvesting, of a generalist superpredator, and of predator cooperation guarantees the emergence of spatial pattern if predator dispersal is sufficiently lower than prey dispersal. Of course, strictly speaking, this conclusion is valid only for the particular functional form used in Table 1 for modelling the p-factor. However, in this specific case, the conclusion should be rather
robust since the functions g in Table 1 are the most standard ones, namely, predator harvesting rate saturating with predator density, generalist superpredator with type II functional response characterized by an attack rate a and a handling time h for the predator population y and by an attack rate a and a handling time h for an extra source of food W (Huxel and McCann, 1998), and predator cooperation positively impacting on predator attack rate. By contrast, factors 4–6 in Table 1, i.e. predator interference (Beddington, 1975), predator intraspecific competition and constant predator stocking, inhibit pattern formation since ∂g/∂y is negative for all parameter values. It is worth noticing that the growth rate g used for the fourth factor, coincides with that proposed in Ruxton (1995) for modelling the effect of prey refuges, so that also antipredator behaviour inhibits pattern formation. Finally, the last three factors, dealing with prey characteristics, do not have any impact on Turing instability because they do not
3452
S. Fasani, S. Rinaldi / Ecological Modelling 222 (2011) 3449–3452
introduce a dependence of the per-capita growth rate g upon predator density. The results reported in Table 1 have been confirmed through the simulation of model (1) on a squared spatial grid with periodic boundary conditions and randomly generated initial conditions. All simulations in cases 4–9 of Table 1 point out spatio-temporal patterns (including travelling waves and vortices) which smoothly vanish as time goes on since in these cases the homogeneous and stationary solution of Eq. (1) is stable. By contrast, in cases 1–3 of Table 1 for positive values of the parameter p and sufficiently hight values of the unbalance k = dx /dy of the dispersal of the two populations the spatio-temporal patterns evolve towards stationary spatial patterns as predicted by our theory. Nine examples of stationary spatial patterns are shown in Fig. 1 where the grey level increases with predator density. More precisely, predator are very abundant (y ≥ 2) in the black regions and practically absent in the white regions. Of course, the opposite holds true for prey, which are, in general, scarce where predator are abundant and vice versa. The spatial patterns reported in the first row of Fig. 1 refer to the first factor considered in Table 1 and are obtained for increasing values of predator attack rate. Similarly, the patterns shown in the second and third rows of Fig. 1 refer to the corresponding rows of Table 1 and are obtained for increasing values of the unbalance k = dx /dy and for increasing values of predator cooperation. The general message emerging from Fig. 1 is that the patterns are, in general, composed of spots, stripes or labyrinths in the abundance of predator (black regions) or in the abundance of prey (white regions). Moreover, the transition from spots to stripes and then labyrinths occurs when a single parameter is varied (the predator attack rate a, the unbalance k and the influence of predator cooperation p in the three rows of Fig. 1). These properties seem to be quite general, because they have been pointed out by all simulations we have performed, as well as by other studies concerning different prey–predator models (see, for example, Banerjee and Petrovskii (2010) for the dependence of the patterns upon predator attack rate). 5. Concluding remarks We have shown in this paper how the emergence of spatial pattern in prey–predator systems can effectively be discussed by using the theory developed long ago by Turing. In particular, we have confirmed that the standard Rosenzweig–MacArthur prey–predator model is critical in the context of Turing theory, so that the addition of any small amount of an extra-factor not considered in that model can easily promote or inhibit pattern formation. The analysis is almost trivial since it requires only to find out if after the introduction of the extra-factor the predator per capita growth rate increases or decreases with predator density. Thus, in many cases, the analysis can be performed on purely intuitive grounds. The nine factors we have considered (see Table 1) are only a few out of many so that some extra-effort would be justified to produce a richer catalog. However, it might be that a factor missing in our list is actually equivalent to one present in the list (like antipredator behaviour due to prey refuges which is equivalent to predator interference). But even in the case that a new factor is really different
from those we have considered in Table 1, it should not be difficult to detect, analytically or numerically, if the predator growth rate increases or decreases with predator density. As a final remark, we like to stress that in this paper we have studied models obtained by slightly perturbing the Rosenzweig–MacArthur model at a stable equilibrium. This implies that the equilibrium of our model is also stable if the newly introduced extra-factor is small and that the prey remains a so-called inhibitor. This means that in the case the extra-factor promotes diffusive instability, the activator is the predator and spatial patterns really emerge if predator disperse sufficiently less than prey. This is not in conflict with studies in which spatial patterns have been detected in cases where prey disperse less than predator, like plant insect communities (Della Rossa et al., 2011). Indeed, in those cases the model is quite different and the equilibrium which is stable in the lumped model and unstable in the spatially extended model has the prey as activator. Acknowledgments The authors are grateful to Egbert van Nes who developed the software grind for spatial simulation and to Fabio Della Rossa and Davide Zambon who performed the numerical analysis. References Alonso, D., Bartumeus, F., Catalan, J., 2002. Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83, 28–34. Banerjee, M., 2010. Self-replication of spatial patterns in a ratio-dependent predator–prey model. Mathematical and Computer Modelling 51, 44–52. Banerjee, M., Petrovskii, S., 2010. Self-organized spatial patterns and chaos in a ratiodependent predator–prey system. Theoretical Ecology. Baurmann, M., Gross, T., Feudel, U., 2007. Instabilities in spatially extended predator–prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. Journal of Theoretical Biology 245, 220–229. Beddington, J., 1975. Mutual interference between parasites or predators. Journal of Animal Ecology 44, 331–340. Chakraborty, A., Singh, M., Lucy, D., Ridland, P., 1996. A numerical study of the formation of spatial patterns in twospotted spider mites. Mathematical and Computer Modelling 49, 1905–1919. Della Rossa, F., Fasani, S., Rinaldi, S., 2011. Potential Turing instability and application to plant-insect models. Submitted for publication. Huxel, G., McCann, K., 1998. Food web stability: the influence of trophic flows across habitats. The American Naturalist 152, 460–469. Levin, S., Segel, L., 1976. Hypothesis for origin of planktonic patchiness. Nature 259, 659. Rosenzweig, M., MacArthur, R., 1963. Graphic representation and stability conditions of predator–prey interaction. The American Naturalist 97, 209–223. Ruxton, G., 1995. Short term refuge use and stability of predator–prey models. Theoretical Population Biology 47, 1–17. Satnoianu, R., Menzinger, M., Maini, P., 2000. Turing instabilities in general systems. Journal of Mathematical Biology 41, 493–512. Segel, L., Jackson, J., 1972. Dissipative structure: an explanation and an ecological example. Journal of Theoretical Biology 37, 545–559. Sun, G., Zhang, G., Jin, Z., 2009. Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dynamics 58, 75–84. Sun, G.Q., Jin, Z., Liu, Q.X., Li, L., 2008. Dynamical complexity of a spatial predator–prey model with migration. Ecological Modelling 219, 248–255. Turing, A., 1952. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 237, 37–72. Wang, W., Liu, Q.X., Jin, Z., 2007. Spatiotemporal complexity of a ratio-dependent predator–prey system. Physics Review E 75, 051913. Wang, W., Zhang, L., Wang, H., Li, Z., 2010. Pattern formation of a predator–prey system with Ivlev-type functional response. Ecological Modelling 221, 131–140. Zhang, L., Wang, W., Xue, Y., 2009. Spatiotemporal complexity of a predator–prey system with constant harvest rate. Chaos, Solitons and Fractals 41, 38–46.
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Short communication
Factors promoting or inhibiting Turing instability in spatially extended prey–predator systems Stefano Fasani a , Sergio Rinaldi a,b,∗ a b
DEI, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy Evolution and Ecology Program, International Institute for Applied Systems Analysis, 2361 Laxenburg, Austria
a r t i c l e
i n f o
Article history: Received 27 January 2011 Received in revised form 29 June 2011 Accepted 3 July 2011 Available online 28 July 2011 Keywords: Prey–predator model Spatial pattern Turing instability Diffusive instability Dispersal
a b s t r a c t The emergence of inhomogeneities in the distributions of the abundances of spatially extended prey–predator systems is investigated. The method of analysis, based on the notion of diffusive (Turing) instability, is systematically applied to nine different models obtained by introducing an extra-factor into the standard Rosenzweig–MacArthur prey–predator model. The analysis confirms that the standard model is critical in the context of Turing instability, and that the introduction of any small amount of the extra-factor can easily promote or inhibit the emergence of spatial patterns. © 2011 Elsevier B.V. All rights reserved.
1. Introduction As it is well known, plant and animal populations are not uniformly distributed in the environment but are packed in patches that remain fixed or move over time (spatio-temporal patterns). The most obvious reasons for patchiness are the occurrence of local perturbations (e.g. the sudden discharge of a chemical species at a specific point) and the presence of permanent spatial inhomogeneities in the environment (e.g. better exposition to sun of one of the river banks). A much more subtle reason for patchiness in ecology has been pointed out long ago by Segel and co-workers (Segel and Jackson, 1972; Levin and Segel, 1976) who used the idea of diffusive instability and the results obtained by Turing (1952) to show that simple prey–predator interactions can be responsible of patchiness even in perfectly homogeneous environments. However, the prey–predator models used by Segel are rather special and not very convincing, in particular in the context of plankton dynamics (Levin and Segel, 1976). Most likely, the reason for this is that more standard models, like the Rosenzweig–MacArthur model (Rosenzweig and MacArthur, 1963), are not appropriate for proving the existence of diffusive instability. In fact, as recognized later in
∗ Corresponding author at: DEI, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy. Tel.: +39 02 2399 3563; fax: +39 02 2399 3412. E-mail address: [email protected] (S. Rinaldi). 0304-3800/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2011.07.002
Alonso et al. (2002), the Rosenzweig–MacArthur model is “critical” in the context of Turing’s theory. More precisely, the two biological mechanisms taken into account in the Rosenzweig–MacArthur model, namely logistic growth of the prey and saturating functional response of the predator, do not imply the emergence of diffusive instability, unless, as shown in this paper, a small amount (actually any small amount) of a suitable extra-mechanism is also included in the model. Since the Rosenzweig–MacArthur model is one of the most popular prey–predator models in ecology, the identification of the extra-mechanisms that must be added to it to obtain a model supporting pattern formation, is an interesting problem. After recalling the basic Turing results and the interpretation given to them in Satnoianu et al. (2000) and in Della Rossa et al. (2011), we show how the problem can be solved with almost no effort. The result of the analysis is that all elementary mechanisms not taken into account in the Rosenzweig–MacArthur model can be partitioned in three classes:
(i) Mechanisms, like predator cooperation or exploitation, promoting diffusive instability if predator disperse sufficiently less than prey. (ii) Mechanisms, like predator interference, intraspecific competition and stocking, always inhibiting diffusive instability. (iii) Mechanisms, like prey harvesting or stocking, with absolutely no influence on diffusive instability.
3450
S. Fasani, S. Rinaldi / Ecological Modelling 222 (2011) 3449–3452
Table 1 Nine independent factors p not taken into account in the Rosenzweig–MacArthur model (4) (first column) and the predator per-capita growth rate g(x, y, p) that should be used in model (2) to take these factors into account (second column). The sign of ∂g/∂y (third column) allows one to identify if the factor p promotes (+) or inhibits (−) diffusive instability. Factor p
Predator per-capita growth rate g(x,y,p)
Sign of
1. Predator harvesting 2. Generalist superpredator
ax e 1+ahx − d − p 1+a1 h y ax 1 e 1+ahx − d − p 1+a h y+a h W
+ +
3. Predator cooperation
a(1+py)x e 1+a(1+py)hx −d
+
4. Predator interference 5. Predator intraspecific competition 6. Predator stocking
ax e 1+ahx+py −d ax e 1+ahx − d − py ax e 1+ahx − d + py
− − −
7. Prey harvesting 8. Prey stocking
ax e 1+ahx −d ax e 1+ahx −d
0 0
a(1−px)x e 1+a(1−px)hx −d
9. Prey cooperation
2. Conditions for diffusive (Turing) instability Spatially extended prey–predator systems can be described, under the assumption of diffusive dispersal, by a PDE of the form ∂x ∂t ∂y ∂t
=
f (x, y, p)x + dx ∇ 2 x
=
g(x, y, p)y + dy ∇ 2 y
(1)
where x and y are prey and predator abundances depending upon time and space in a given domain of R2 , f and g are per-capita growth rates depending upon demographic parameters p, and dx and dy are the non-negative dispersal coefficients of the two populations. Typically, in order to have a well posed problem, zero-flux or periodic boundary conditions are also imposed. ¯ of Eq. (1) A positive, homogeneous and stationary solution (¯x, y) (characterized by f = g = 0) can be stable in absence of diffusion (i.e. for dx = dy = 0) but unstable for suitable pairs (dx ,dy ). In other words, ¯ can be stable in a lumped prey–predator model an equilibrium (¯x, y) dx dt dy dt
=
f (x, y, p)x
=
g(x, y, p)y
(2)
but unstable in its spatially extended version Eq. (1). This somehow counterintuitive phenomenon, first investigated in a celebrated paper of Turing (1952), is known as diffusion-induced instability, but is also called Turing instability. The original Turing analysis considers the effect of small pertur¯ in the Fourier expansion of the solution bations imposed on (¯x, y) of Eq. (1) and the key result (Satnoianu et al., 2000) is that diffusion induced instability is equivalent to the instability of the matrix C = A(p) − D
(3)
¯ i.e. where A(p) is the Jacobian of Eq. (2) at the equilibrium (¯x, y),
⎡ ∂f x¯ A(p) = ⎣ ∂x y¯
∂g ∂x
∂f ∂y ∂g y¯ ∂y x¯
⎤ ⎦
and D is the diagonal matrix with dx and dy as diagonal elements. If dx = dy , i.e. if D is proportional to the identity matrix, the spectrum of C in Eq. (3) is simply the spectrum of A shifted to the left, so that C cannot be unstable if A is stable. This is why the dispersion coefficients must be unbalanced in order to have Turing instability. As shown in (Della Rossa et al., 2011; Satnoianu et al., 2000), the ¯ is stable in Eq. problem of finding triplets (p,dx ,dy ) for which (¯x, y) (2) but unstable in Eq. (1) can be solved in two steps, namely: (i) find values of p for which one diagonal element of the Jacobian matrix A is positive. Notice that if aii is positive (i.e. if i is the
∂g ∂y
0
/ i, is negative, since A is stable so-called activator) then ajj , j = in Eq. (2) so that tr(A) must be negative. (ii) determine the dispersal coefficients (dx , dy ) realizing Turing instability by imposing that C is unstable. As shown in Satnoianu et al. (2000), this is always possible if the activator disperses sufficiently less than the other population (inhibitor). This decomposition, which has never been systematically exploited in the literature (see Segel and Jackson (1972), Levin and Segel (1976), Chakraborty et al. (1996), Alonso et al. (2002), Baurmann et al. (2007), Wang et al. (2007), Banerjee (2010), Banerjee and Petrovskii (2010), Sun et al. (2009), Sun et al. (2008), Wang et al. (2010), Zhang et al. (2009)), greatly simplifies the analysis, because the two steps (i) and (ii) can be easily accomplished. Moreover, step (ii) is often not needed in ecology, where the problem of major concern is the identification of the factors promoting or inhibiting diffusion-induced instability. 3. Criticality of the Rosenzweig–MacArthur model One of the most standard prey–predator models (Rosenzweig and MacArthur, 1963) is composed of a logistic prey and a type II predator and is therefore described by
x ay dx = r 1− − x K dt 1 + ahx dy ax = e −d y dt 1 + ahx
(4)
where r and K are prey net growth rate and carrying capacity and a, h, e and d are predator attack rate, handling time, efficiency and death rate, respectively. Model (4) in the form of Eq. (2) and ¯ for suitable values of its has a positive stable equilibrium (¯x, y) parameters. Since the per-capita growth rate g does not depend on predator density, the second diagonal element of the Jacobian A in Eq. (3) is zero, i.e. a22 = 0 for all parameter values, while a11 < 0 since tr(A) < 0 at a stable equilibrium. This means that Turing condition (a11 a22 < 0) is not satisfied in model (4), since in that model a11 a22 = 0. In other words, the Rosenzweig–MacArthur model is critical in the context of Turing instability. 4. Small factors can trigger Turing instability The addition to model (4) of any small amount p of an extra ¯ as well demographic factor slightly modifies the equilibrium (¯x, y) as the elements a11 and a22 of the matrix A. Hence, by continuity, the ¯ remains positive and stable and a11 remains negequilibrium (¯x, y) ative. By contrast, depending upon the p-factor, a22 , which is zero for p = 0, can be positive or negative (or equal to zero) for small values of p. In the first case (a22 (p) > 0 for small values of p > 0) the p-factor promotes Turing instability,i.e. pattern formation, if
S. Fasani, S. Rinaldi / Ecological Modelling 222 (2011) 3449–3452
3451
Fig. 1. Stationary spatial patterns of predator density obtained through simulation of model (1) on a squared spatial grid with periodic boundary conditions. First row: factor 1 in Table 1. Second row: factor 2 in Table 1. Third row: factor 3 in Table 1. Common reference parameter values: r = K = e = 1, h = 3, d = 0.2. Other parameter values: first row, p = 0.09, dx = 0.005, a = h = 1, k = 1000, a = 0.95,1,1.05 from left to right. Second row: p = 0.1, dx = 0.005, a = a = a = 1, h = h = 3, W = 0.05, k = 500,1000,1500 from left to right. Third row: k = 10,000, dx = 0.01, a = 0.9, p = 0.20,0.22,0.24 from left to right.
predator disperse sufficiently less than prey. The conclusion is that the factors promoting Turing instability can be simply identified by looking at the sign of ∂g/∂y at equilibrium. We now consider nine independent factors that are not taken into account in model (4) but can be easily added to it by modifying the predator per capita growth rate. Table 1 reports the nine factors, the functions g(x,y,p) and the signs of ∂g/∂y. Of course g(x, y, 0) is the Rosenzweig–MacArthur predator growth rate in all cases. For the first three factors, ∂g/∂y is positive for all parameter values, so that one can conclude that even the existence of a small amount of predator harvesting, of a generalist superpredator, and of predator cooperation guarantees the emergence of spatial pattern if predator dispersal is sufficiently lower than prey dispersal. Of course, strictly speaking, this conclusion is valid only for the particular functional form used in Table 1 for modelling the p-factor. However, in this specific case, the conclusion should be rather
robust since the functions g in Table 1 are the most standard ones, namely, predator harvesting rate saturating with predator density, generalist superpredator with type II functional response characterized by an attack rate a and a handling time h for the predator population y and by an attack rate a and a handling time h for an extra source of food W (Huxel and McCann, 1998), and predator cooperation positively impacting on predator attack rate. By contrast, factors 4–6 in Table 1, i.e. predator interference (Beddington, 1975), predator intraspecific competition and constant predator stocking, inhibit pattern formation since ∂g/∂y is negative for all parameter values. It is worth noticing that the growth rate g used for the fourth factor, coincides with that proposed in Ruxton (1995) for modelling the effect of prey refuges, so that also antipredator behaviour inhibits pattern formation. Finally, the last three factors, dealing with prey characteristics, do not have any impact on Turing instability because they do not
3452
S. Fasani, S. Rinaldi / Ecological Modelling 222 (2011) 3449–3452
introduce a dependence of the per-capita growth rate g upon predator density. The results reported in Table 1 have been confirmed through the simulation of model (1) on a squared spatial grid with periodic boundary conditions and randomly generated initial conditions. All simulations in cases 4–9 of Table 1 point out spatio-temporal patterns (including travelling waves and vortices) which smoothly vanish as time goes on since in these cases the homogeneous and stationary solution of Eq. (1) is stable. By contrast, in cases 1–3 of Table 1 for positive values of the parameter p and sufficiently hight values of the unbalance k = dx /dy of the dispersal of the two populations the spatio-temporal patterns evolve towards stationary spatial patterns as predicted by our theory. Nine examples of stationary spatial patterns are shown in Fig. 1 where the grey level increases with predator density. More precisely, predator are very abundant (y ≥ 2) in the black regions and practically absent in the white regions. Of course, the opposite holds true for prey, which are, in general, scarce where predator are abundant and vice versa. The spatial patterns reported in the first row of Fig. 1 refer to the first factor considered in Table 1 and are obtained for increasing values of predator attack rate. Similarly, the patterns shown in the second and third rows of Fig. 1 refer to the corresponding rows of Table 1 and are obtained for increasing values of the unbalance k = dx /dy and for increasing values of predator cooperation. The general message emerging from Fig. 1 is that the patterns are, in general, composed of spots, stripes or labyrinths in the abundance of predator (black regions) or in the abundance of prey (white regions). Moreover, the transition from spots to stripes and then labyrinths occurs when a single parameter is varied (the predator attack rate a, the unbalance k and the influence of predator cooperation p in the three rows of Fig. 1). These properties seem to be quite general, because they have been pointed out by all simulations we have performed, as well as by other studies concerning different prey–predator models (see, for example, Banerjee and Petrovskii (2010) for the dependence of the patterns upon predator attack rate). 5. Concluding remarks We have shown in this paper how the emergence of spatial pattern in prey–predator systems can effectively be discussed by using the theory developed long ago by Turing. In particular, we have confirmed that the standard Rosenzweig–MacArthur prey–predator model is critical in the context of Turing theory, so that the addition of any small amount of an extra-factor not considered in that model can easily promote or inhibit pattern formation. The analysis is almost trivial since it requires only to find out if after the introduction of the extra-factor the predator per capita growth rate increases or decreases with predator density. Thus, in many cases, the analysis can be performed on purely intuitive grounds. The nine factors we have considered (see Table 1) are only a few out of many so that some extra-effort would be justified to produce a richer catalog. However, it might be that a factor missing in our list is actually equivalent to one present in the list (like antipredator behaviour due to prey refuges which is equivalent to predator interference). But even in the case that a new factor is really different
from those we have considered in Table 1, it should not be difficult to detect, analytically or numerically, if the predator growth rate increases or decreases with predator density. As a final remark, we like to stress that in this paper we have studied models obtained by slightly perturbing the Rosenzweig–MacArthur model at a stable equilibrium. This implies that the equilibrium of our model is also stable if the newly introduced extra-factor is small and that the prey remains a so-called inhibitor. This means that in the case the extra-factor promotes diffusive instability, the activator is the predator and spatial patterns really emerge if predator disperse sufficiently less than prey. This is not in conflict with studies in which spatial patterns have been detected in cases where prey disperse less than predator, like plant insect communities (Della Rossa et al., 2011). Indeed, in those cases the model is quite different and the equilibrium which is stable in the lumped model and unstable in the spatially extended model has the prey as activator. Acknowledgments The authors are grateful to Egbert van Nes who developed the software grind for spatial simulation and to Fabio Della Rossa and Davide Zambon who performed the numerical analysis. References Alonso, D., Bartumeus, F., Catalan, J., 2002. Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83, 28–34. Banerjee, M., 2010. Self-replication of spatial patterns in a ratio-dependent predator–prey model. Mathematical and Computer Modelling 51, 44–52. Banerjee, M., Petrovskii, S., 2010. Self-organized spatial patterns and chaos in a ratiodependent predator–prey system. Theoretical Ecology. Baurmann, M., Gross, T., Feudel, U., 2007. Instabilities in spatially extended predator–prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. Journal of Theoretical Biology 245, 220–229. Beddington, J., 1975. Mutual interference between parasites or predators. Journal of Animal Ecology 44, 331–340. Chakraborty, A., Singh, M., Lucy, D., Ridland, P., 1996. A numerical study of the formation of spatial patterns in twospotted spider mites. Mathematical and Computer Modelling 49, 1905–1919. Della Rossa, F., Fasani, S., Rinaldi, S., 2011. Potential Turing instability and application to plant-insect models. Submitted for publication. Huxel, G., McCann, K., 1998. Food web stability: the influence of trophic flows across habitats. The American Naturalist 152, 460–469. Levin, S., Segel, L., 1976. Hypothesis for origin of planktonic patchiness. Nature 259, 659. Rosenzweig, M., MacArthur, R., 1963. Graphic representation and stability conditions of predator–prey interaction. The American Naturalist 97, 209–223. Ruxton, G., 1995. Short term refuge use and stability of predator–prey models. Theoretical Population Biology 47, 1–17. Satnoianu, R., Menzinger, M., Maini, P., 2000. Turing instabilities in general systems. Journal of Mathematical Biology 41, 493–512. Segel, L., Jackson, J., 1972. Dissipative structure: an explanation and an ecological example. Journal of Theoretical Biology 37, 545–559. Sun, G., Zhang, G., Jin, Z., 2009. Predator cannibalism can give rise to regular spatial pattern in a predator–prey system. Nonlinear Dynamics 58, 75–84. Sun, G.Q., Jin, Z., Liu, Q.X., Li, L., 2008. Dynamical complexity of a spatial predator–prey model with migration. Ecological Modelling 219, 248–255. Turing, A., 1952. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences 237, 37–72. Wang, W., Liu, Q.X., Jin, Z., 2007. Spatiotemporal complexity of a ratio-dependent predator–prey system. Physics Review E 75, 051913. Wang, W., Zhang, L., Wang, H., Li, Z., 2010. Pattern formation of a predator–prey system with Ivlev-type functional response. Ecological Modelling 221, 131–140. Zhang, L., Wang, W., Xue, Y., 2009. Spatiotemporal complexity of a predator–prey system with constant harvest rate. Chaos, Solitons and Fractals 41, 38–46.