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Competition, predation and coexistence in a three trophic system
Ecological Modelling 220 (2009) 2349–2352
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Competition, predation and coexistence in a three trophic system M. Florencia Carusela a,b,∗ , Fernando R. Momo a , Lilia Romanelli a,b a b
Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M.Gutiérrez 1150, Los Polvorines, Buenos Aires, Argentina CONICET, Argentina
a r t i c l e
i n f o
Article history: Received 12 January 2009 Received in revised form 22 May 2009 Accepted 2 June 2009 Available online 2 July 2009 Keywords: Population dynamics Continuous models Trophic systems Amphipods
a b s t r a c t Simple ecological models that mostly operate with population densities using continuous variables, explain quite well the behavior of real populations. In this work we propose and discuss the continuous dynamics of a system of three species, which belongs to the well-known family of Lotka–Volterra models. In particular, the proposed model includes direct effects such as predation and competition among species, and indirect effects such as refuge. The model is proposed to explain recent studies about a group of crustacean (amphipods of genus Hyallela) found in all the plain streams and shallow lakes of the American continent. The studied system includes three compartments: algae, a strictly herbivore amphipod and an omnivore (herbivore and carnivore) one. The analysis of the model shows that there are stable extinction equilibria throughout all the parameters’ space. There are also equilibria with stable coexistence of the three species and two interesting binary equilibria: one with stable coexistence of algae and herbivore and other with coexistence between algae and omnivore amphipods. The presence of Allee effect in the algae growth and the existence of refuge for the herbivore amphipod (prey) determine a bottom-up control. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Most of the studied systems in population ecology are binary systems; however, there are several models that involucrate three species. These models are so called tritrophic systems and may show complex dynamics with limit cycles or even chaos. Maionchi et al. (2006) for instance, present a tritrophic model that exhibits chaos and complex spatial patterns. Ramos-Jiliberto et al. (2008) developed a model where inducible defenses can stabilize the system dynamics. Predators’ functional responses are very important in order to determine the dynamic behavior of these models (see for instance Naji and Balasim, 2007); predators’ efficiency is also a characteristic key to analyze the tritrophic systems stability (Cassinari et al., 2007). Regarding to the complex web of trophic relationships in nature, at the lowest trophic level the resources are abiotic (light, nutrients, etc.). But organisms at all other levels, except the top level, are simultaneously both consumers of resources below and resource for the consumers above (Getz, 1999). If an organism is an omnivore, it can eat from two consecutive trophic levels and so, if it exploits a prey and the prey of that prey, it becomes a predator and a competitor of the same species simultaneously. Probably this sit-
∗ Corresponding author at: Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M.Gutiérrez 1150, Los Polvorines, Buenos Aires, Argentina. E-mail address: fl[email protected] (M. Florencia Carusela). 0304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2009.06.008
uation is not rare because when a species is omnivore, frequently eats adjacent trophic levels (Pimm, 1999). An interesting example of a three trophic systems can be found in freshwater amphipods (Crustacea) of the genus Hyalella; some species of this genus (for example Hyallella curvispina) can combine herbivore and carnivore diets, depending on the offer of algae food (Casset et al., 2001); another species (i.e. Hyalella pseudoazteca) are strictly herbivore. The consequence of that food behavior is a complex set of interactions that combines competition and predation between the same couple of populations. In the Pampean Region of Argentina (34–40◦ South, 58–63◦ West) H. curvispina and H. pseudoazteca are the two most frequent amphipod species; both species are spread in all South-America. These species form an ecological assemblage together with benthic microalgae and submersed macrophytes (mainly Myriophyllum sp., Ceratophyllum sp. and Egeria sp.). Algae give food resources for amphipods, and macrophytes give refuge for the crustaceans and a large substratum where algae can grow. Amphipods eat mainly algae biofilms. Casset et al. (2001) have described the population dynamics of the two amphipod species, H. curvispina and H. pseudoazteca, that inhabit together in streams across all South America, from Amazonian to Patagonian (González, 2003; González and Watling, 2003). In their paper, Casset et al. (2001) examined abundances and biomass of amphipods along a year in a plain stream from the Pampean region of Argentina. The study was carried out in a low velocity stream, and recorded amphipods abundance and microalgae biomass. In that study, it was found that H. curvispina
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M. Florencia Carusela et al. / Ecological Modelling 220 (2009) 2349–2352
where a set of non-dimensional variables are defined as follows: x=
A , K
y=
N1 , K
z=
N2 K
r1 r2 KP1 , R2 = , H1 = , m q Q = , M = , t = T K K
R1 =
Fig. 1. The model. The biological compartments are indicated with boxes and the interactions among them with directional arrows. N1 , N2 and A indicate the biomass of H. pseudoazteca, H. curvispina and algae, respectively.
eats H. pseudoazteca individuals when the algae food is scarce (Giorgi and Tiraboschi, 1999; Casset et al., 2001); in consequence, H. curvispina acts as an omnivore species. Because H. pseudoazteca predates on the same algae that H. curvispina, the relationship between these two species is a mixture of predation and competition occurring simultaneously at different intensities. We ask ourselves if the system formed by the two Hyallela species is dynamically stable. To answer this question, it is necessary to study the behavior of the system under different conditions and to consider direct effects (predation and competition) and indirect ones (the refuge that macrophytes give to the prey amphipod). In this paper, we present a model for this system based on Lotka–Volterra type equations with combined interactions. This approach is the first to focus the problem of the combination of roles between competition and predation in the same species.
2. The model The biological system consists of three compartments (populations): algae (A), H. pseudoazteca (N1 ) and H. curvispina (N2 ). Biomass of each compartment has inputs and outputs that depend on both intrinsic and environmental variables. This system can be sketched as in Fig. 1. For H. pseudoazteca (N1 ) algae food and the refuge (given by macrophytes) are inflows, and mortality and predation are the outflows. For H. curvispina (N2 ) inputs come from predation on algae and amphipods, and the output is mortality. The model can be written as a system of ODE since it is proposed to represent the dynamics in a pool, a part of a stream. It is accepted that plain streams work as a sort of linked pools (backwaters) connected by rapids (Feijoo et al., 1999). So it can be considered that each pool has its proper populations, which have a local dynamics. The scheme shown in Fig. 1 can be expressed by a system of three coupled dimensionless equations of the form: dx = x(1 − x) dt
x M
− 1 − H1 xy − H2 xz
dy = −R1 y + e1 H1 xy − dt
dz = −R2 z + e2 H2 xz + e3 dt
KP2 ,
H3 =
KP3
(4)
where A = algae biomass (kg/m2 ), N1 = H. pseudoazteca biomass (kg/m2 ), N2 = H. curvispina biomass (kg/m2 ); e1 , e2 , and e3 are the parameters which corresponds to the assimilation efficiencies (0 < ei < 1), K is the carrying capacity of algae (kg/m2 ), is the growth rate of algae (day−1 ); P1 , P2 and P3 are predation efficiencies (day−1 kg−1 m−2 ); r1 and r2 are the growth rates of amphipods (day−1 ) which are negative in absence of preys (the signs are made explicit in the equations so that all the constants are positive, like in Lotka–Volterra type models). Parameter q denotes the number of refugee preys; m is the critical density of algae (threshold) needed to start to grow, according to previous models of periphytic and microbenthic algae dynamics (Momo, 1995; Saravia et al., 1999). M is the threshold expressed as a fraction of the carrying capacity of the algae K. Finally, t is the time but normalized to the intrinsic grow rate of the algae (t = T, T in units of days). The first term in Eq. (1) describes the monotonic changing of populations size with strong Allee effect for algae; this is a common observed phenomenon in microalgae (Momo, 1995; Saravia et al., 1999). The last two terms in Eq. (1) represent predation due to both species of Hyallela, with non-dimensional coefficients of predation H1 and H2 . H. pseudoazteca can be predated by H. curvispina but H. pseudoazteca can use macrophytes like refuge, showing a demographic barrier to predation because a fix stock Q of prey population individuals cannot be predated due to they are refugee. Macrophytes (refuge) have a dynamics several orders of magnitude slower than the dynamics of invertebrate and microalgae. In consequence, in periods of hydraulic stability (without floods), we can assume that macrophyte biomass remains roughly constant, therefore the number of refugee amphipods Q can also be considered constant. On the contrary, in that period (usually several months) amphipods and algae have an intense demography dynamics. The assumption Q = cte. is represented in the last terms of Eqs. (2) and (3). As a first approach to the model, we made several simplifications: the parameter H3 , the coefficient of predation of H. curvispina over H. pseudoazteca is considered a constant. Probably this assumption is not realistic because H3 should be a function of the density of algae (x); however, we made this approximation in order to simplify the dynamical analysis. The study of the model with non-constant H3 is the object of a future work. In the same way, we regard parameters e1 , e2 , e3 (conversion efficiencies) as constants since they are physiological traits of populations that can be considered as genetically determined; conversely, R1 and R2 are parameters that are strongly influenced by the habitat quality. Therefore, we consider these last two parameters to define the parameter space for our model.
(1)
H3 (y − Q )z
y≥Q
0
y
H2 =
H3 (y − Q )z
y≥Q
0
y
3. Results (2)
(3)
3.1. Analyzing the dynamics In order to make an outline of our problem, we obtain the fixed points of the system given in Eqs. (1)–(3). The fixed points are solutions of dx/dt = dy/dt = dz/dt = 0, that is the intersection of the
M. Florencia Carusela et al. / Ecological Modelling 220 (2009) 2349–2352
2351
nullclines of the system given by dx = 0 ⇔ x = 0 ∨ (1 − x) dt
x M
dy = 0 ⇔ −R1 y + e1 H1 xy − dt
− 1 − H1 y − H2 z = 0
H3 (y − Q )z
y≥Q
0
y
dz = 0 ⇔ z = 0 ∨ −R2 + e2 H2 x + e3 dt
(5)
=0
H3 (y − Q )
y≥Q
0
y
(6)
=0
(7)
The factor (y − Q) in the third term in Eqs. (6) and (7) represents the number of individuals y that is available to the predator z, in consequence it must be kept positive during the whole evolution. Therefore, when this term becomes negative, it is switched to zero (there are no available preys). From Eqs. (5) and (7) it can be seen that the yz and the xy planes are nullcline planes for x and z, respectively; as is the xz plane for y, when y < Q. From Eqs. (5)–(7), different equilibrium points Wi are obtained which represent different biological situations: W1 = (0, 0, 0). Extinction of the three species. W2 = (1, 0, 0). Only algae in its carrying capacity without amphipods. W3 = (M, 0, 0). Only algae in its threshold density. W4 = (x4 , y4 , 0) with 0 < x4 , y4 . Binary system without the omnivore amphipod. W5 = (x5 , 0, z5 ) with 0 < x5 , z5 . Binary system without the herbivore amphipod. W6 = (x6 , y6 , z6 ) with 0 < z6 < y6 ; 0 < x6 < y6 and 0 < Q < y6 . Tritrophic system; the herbivore amphipod is more abundant than the omnivore one. W7 = (x7 , y7 , z7 ) with 0 < Q < y7 , x7 < z7 and y7 < z7 . Tritrophic system; the omnivore amphipod is more abundant than the herbivore one. 3.2. Stability analysis and parameters The set of parameters in our model is divided in two groups. The first one constitutes the parameters that remain quite constant in the time scales considered. The assimilation efficiencies ei are strongly determined by physiology restrictions and genetics; the Pi ’s depend on morphologic features of the amphipods. Therefore, these parameters are included in the first subset. On the other hand, the mortality and growth rates have a strong variability in field due to their dependence on environmental conditions, hydrological conditions, etc. These parameters are included in the second subset. According to these considerations, we study numerically the dynamical response of the system defining our parameter space as a function of R1 and R2 . From Casset et al. (2001), we are aware that the mortality rates of amphipods are quite smaller than the growth rate of the algae. Therefore we propose values and ranges for the parameters as follows: Ri < 1,
Hi 1,
Q 1,
M 1,
ei 1
with
i = 1, 2
H3 < H1 , H2 , e3 < e1 , e2 e1 = 0.02,
e2 = 0.03,
e3 = 0.009,
H3 = 0.07,
H1 = 1.5,
H2 = 1.
M = 0.02,
Q = 0.01,
The values of the parameters were obtained from several field studies carried out in the Luján river basin (Buenos Aires, Argentina) and some laboratory experiments. Original data can be consulted in Giorgi and Malacalza (1994), Saravia et al. (1998, 1999), Giorgi et al. (1998) and Giorgi and Ferreyra (2000).
Fig. 2. Regions of stable solutions in the parameter space (R1 vs. R2 ). The different zones are discussed in the text. The values of the constant parameters are: e1 = 0.02, e2 = 0.03, e3 = 0.009, M = 0.02, Q = 0.01, H3 = 0.07, H1 = 1.5, H2 = 1. The numerical calculations are obtained for a grid of 40 × 40 values.
Defining our parameter space as R1 , R2 ∈ [0, 0.03], we can characterize the equilibrium points mentioned above, taking into account their biological meaning. Analyzing eigenvalues for the linearized system around a fixed point, we study the stability of the solutions Wi (Guckenheimer and Holmes, 1983; Momo and Capurro, 2006), finding stable and unstable solutions. The stable ones are presented in Fig. 2. W1 → stable solution in the part of the parameter space corresponding to the model: Regions I–II–III–IV (three real negative eigenvalues, atractor) W2 → unstable solution in the part of the parameter space corresponding to the model (two positive and one negative real eigenvalues, saddle node) W3 → unstable solution in the part of the parameter space corresponding to the model (one positive and two negative real eigenvalues, saddle node) W4 → stable solution in Region IV (three negative real eigenvalues, atractor) W5 → stable solution in Region III (three negative real eigenvalues, atractor) W6 → unstable solution in the part of the parameter space corresponding to the model W7 → stable solution in Regions II–III–IV (three negative real eigenvalues, atractor). Fig. 2 shows a summary of the cases previously discussed. The parameter space can be divided in several zones, where each one corresponds to one of the following coexistence of solutions: Zone I → extinction of three species (W1 ). Zone II → stable coexistence of the three species (W7 ) and extinction of three species (W1 ). Zone III → stable coexistence of algae and H. curvispina (W5 ), stable coexistence of algae, H. pseudoazteca and H. curvispina (W7 ) and extinction of three species (W1 ). Zone IV → stable coexistence of algae, H. pseudoazteca and H. curvispina (W7 ), stable coexistence of algae and H. pseudoazteca (W4 ) and extinction of three species (W1 ). 4. Discussion Throughout in all the parameter space it is possible to find a stable situation with the absence (extinction) of the three species. This is due by the strong Allee effect occurring in the algae growth. Effectively, algae biomass needs to reach a threshold (M) to have
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a positive growth rate and without algae, there is not support for amphipods populations. Another interesting behavior is found in zone II. In this region, the omnivore species has growth rates with high absolute values and, independently of the growth rate of the herbivore amphipod, there is the possibility of the stable coexistence of the three species. This is because the H. curvispina population decays faster than the H. pseudazteca one, avoiding the extinction of the prey amphipod. However, this occurs when the biomass of H. curvispina is higher than the biomass of the herbivore H. pseudoazteca, which means that H. curvispina is processing the highest fraction of the available energy and it is harming the other amphipod by competition as well as predation. This fact explains why in Pampean streams it is frequent to find higher biomass of H. curvispina whereas H. pseudoazteca is much less abundant (Casset et al., 2001). The coexistence of algae and only one of the amphipods species is restricted to zones III and IV. At zone III, when the growth rate of the herbivore is higher (in absolute value) than the omnivore one, the stable binary coexistence is between algae and the omnivore because the herbivore amphipod population decays at higher velocity than the other amphipod species. On the other hand, at zone IV, the reciprocal situation is stable (algae and herbivore) despite y > Q, so there are available preys for H. curvispina; the reason is again the difference between both net growth rate that, in this case, favors to the herbivore amphipod. It is important to mention that the behavior in each zone is highly dependent of the initial conditions. Further studies must be oriented to examine the size of the attractive basin for each equilibrium point or the effect of other population parameters, like efficiencies (Cassinari et al., 2007), that would determine chaotic behavior and bifurcation points. In our system, there is a strong bottom-up control given by the growth threshold of the algae. Moreover, we can consider that in the three species coexistence case, the ecological control can be characterized as top-down; this is only if we accept that the control can be characterized by the stocks of each species determining the amount of energy that flux throughout them. An exhaustive analysis of the energy fluxes would required another type of model since this one cannot allows us to analyze fluxes.
Acknowledgment This work was partially supported by CONICET (PIP 6124). References Casset, M., Momo, F., Giorgi, A., 2001. Dinámica poblacional de dos especies de anfípodos y su relación con la vegetación acuática en un microambiente de la cuenca del río Luján (Argentina). Ecología Austral 11, 79–85. Cassinari, M.P., Groppi, M., Tebaldi, C., 2007. Effects of predation efficiencies on the dynamics of a tritrophic food chain. Math. Biosci. Eng. 4, 431–456. Feijoo, C.S., Giorgi, A.D.N., García, M.E., Momo, F.R., 1999. Temporal and spatial variability in streams of a Pampean basin. Hydrobiologia 394, 41–52. Getz, W.M., 1999. Population and evolutionary dynamics of consumer–resource systems. In: McGlade, J. (Ed.), Advanced Ecological Theory. Principles and Applications. Blackwell Science, Oxford, pp. 194–231. Giorgi, A., Malacalza, L., 1994. Biomass variation of microphytobenthos in a plain stream. Verh. Internat. Verein. Limnol. 25, 1883–1887. ˜ P., Duttweiler, F., 1998. Annual variation of periphyGiorgi, A., Feijoó, C., Calvino, ton biomass in two plain streams with different macrophyte abundance. Verh. Internat. Verein. Limnol. 26, 1698–1701. Giorgi, A.D.N., Tiraboschi, B., 1999. Evaluación experimental del efecto de dos grupos de macroinvertebrados (anfípodos y gasterópodos) sobre algas epífitas. Ecología Austral 9 (1), 35–44. Giorgi, A., Ferreyra, G., 2000. Phytobenthos colonization in a lowland stream in Argentina. J. Freshwater Ecol. 15 (1), 39–46. González, E., 2003. The freshwater amphipods Hyalella Smith, 1874 in Chile (Crustacea: Amphipoda). Revista Chilena de Historia Natural 76, 623–637. González, E., Watling, L., 2003. A new species of Hyalella from Brazil (Crustacea: Amphipoda: Hyalellidae), with redescriptions of three other species in the genus. J. Nat. Hist. 37, 2045–2076. Guckenheimer, J., Holmes, P., 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York. Maionchi, D.O., dos Reis, S.F., Aguiar, M.A.M., 2006. Chaos and pattern formation in a spatial tritrophic food chain. Ecol. Model. 191 (2), 291–303. Momo, F.R., 1995. A new model for periphyton growth in running waters. Hydrobiologia 299, 215–218. Momo, F., Capurro, A., 2006. Ecología Matemática, Principios y Aplicaciones, Ediciones Cooperativas (and references therein). Naji, R.K., Balasim, A.T., 2007. Dynamical behavior of a three species food chain model with Beddington–DeAngelis functional response. Chaos Solitons Fract. 32 (5), 1853–1866. Pimm, S.L., 1999. The dynamics of the flows of matter and energy. In: McGlade, J. (Ed.), Advanced Ecological Theory. Principles and Applications. Blackwell Science, Oxford, pp. 172–193. Ramos-Jiliberto, R., Mena-Lorca, J., Flores, J.D., Morales-Álvarez, W., 2008. Role of inducible defenses in the stability of a tritrophic system. Ecol. Complex. 5 (2), 183–192. Saravia, L.A., Momo, F.R., Boffi Lissin, L.D., 1998. Modelling periphytons’s dynamics in ruinning waters. Ecol. Model. 114, 35–47. Saravia, Giorgi, Momo, 1999. A photographic method for periphyton chlorophyll-a estimation in artificial sustrata. Aquat. Ecol. 33, 325–330.
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Competition, predation and coexistence in a three trophic system M. Florencia Carusela a,b,∗ , Fernando R. Momo a , Lilia Romanelli a,b a b
Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M.Gutiérrez 1150, Los Polvorines, Buenos Aires, Argentina CONICET, Argentina
a r t i c l e
i n f o
Article history: Received 12 January 2009 Received in revised form 22 May 2009 Accepted 2 June 2009 Available online 2 July 2009 Keywords: Population dynamics Continuous models Trophic systems Amphipods
a b s t r a c t Simple ecological models that mostly operate with population densities using continuous variables, explain quite well the behavior of real populations. In this work we propose and discuss the continuous dynamics of a system of three species, which belongs to the well-known family of Lotka–Volterra models. In particular, the proposed model includes direct effects such as predation and competition among species, and indirect effects such as refuge. The model is proposed to explain recent studies about a group of crustacean (amphipods of genus Hyallela) found in all the plain streams and shallow lakes of the American continent. The studied system includes three compartments: algae, a strictly herbivore amphipod and an omnivore (herbivore and carnivore) one. The analysis of the model shows that there are stable extinction equilibria throughout all the parameters’ space. There are also equilibria with stable coexistence of the three species and two interesting binary equilibria: one with stable coexistence of algae and herbivore and other with coexistence between algae and omnivore amphipods. The presence of Allee effect in the algae growth and the existence of refuge for the herbivore amphipod (prey) determine a bottom-up control. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Most of the studied systems in population ecology are binary systems; however, there are several models that involucrate three species. These models are so called tritrophic systems and may show complex dynamics with limit cycles or even chaos. Maionchi et al. (2006) for instance, present a tritrophic model that exhibits chaos and complex spatial patterns. Ramos-Jiliberto et al. (2008) developed a model where inducible defenses can stabilize the system dynamics. Predators’ functional responses are very important in order to determine the dynamic behavior of these models (see for instance Naji and Balasim, 2007); predators’ efficiency is also a characteristic key to analyze the tritrophic systems stability (Cassinari et al., 2007). Regarding to the complex web of trophic relationships in nature, at the lowest trophic level the resources are abiotic (light, nutrients, etc.). But organisms at all other levels, except the top level, are simultaneously both consumers of resources below and resource for the consumers above (Getz, 1999). If an organism is an omnivore, it can eat from two consecutive trophic levels and so, if it exploits a prey and the prey of that prey, it becomes a predator and a competitor of the same species simultaneously. Probably this sit-
∗ Corresponding author at: Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M.Gutiérrez 1150, Los Polvorines, Buenos Aires, Argentina. E-mail address: fl[email protected] (M. Florencia Carusela). 0304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2009.06.008
uation is not rare because when a species is omnivore, frequently eats adjacent trophic levels (Pimm, 1999). An interesting example of a three trophic systems can be found in freshwater amphipods (Crustacea) of the genus Hyalella; some species of this genus (for example Hyallella curvispina) can combine herbivore and carnivore diets, depending on the offer of algae food (Casset et al., 2001); another species (i.e. Hyalella pseudoazteca) are strictly herbivore. The consequence of that food behavior is a complex set of interactions that combines competition and predation between the same couple of populations. In the Pampean Region of Argentina (34–40◦ South, 58–63◦ West) H. curvispina and H. pseudoazteca are the two most frequent amphipod species; both species are spread in all South-America. These species form an ecological assemblage together with benthic microalgae and submersed macrophytes (mainly Myriophyllum sp., Ceratophyllum sp. and Egeria sp.). Algae give food resources for amphipods, and macrophytes give refuge for the crustaceans and a large substratum where algae can grow. Amphipods eat mainly algae biofilms. Casset et al. (2001) have described the population dynamics of the two amphipod species, H. curvispina and H. pseudoazteca, that inhabit together in streams across all South America, from Amazonian to Patagonian (González, 2003; González and Watling, 2003). In their paper, Casset et al. (2001) examined abundances and biomass of amphipods along a year in a plain stream from the Pampean region of Argentina. The study was carried out in a low velocity stream, and recorded amphipods abundance and microalgae biomass. In that study, it was found that H. curvispina
2350
M. Florencia Carusela et al. / Ecological Modelling 220 (2009) 2349–2352
where a set of non-dimensional variables are defined as follows: x=
A , K
y=
N1 , K
z=
N2 K
r1 r2 KP1 , R2 = , H1 = , m q Q = , M = , t = T K K
R1 =
Fig. 1. The model. The biological compartments are indicated with boxes and the interactions among them with directional arrows. N1 , N2 and A indicate the biomass of H. pseudoazteca, H. curvispina and algae, respectively.
eats H. pseudoazteca individuals when the algae food is scarce (Giorgi and Tiraboschi, 1999; Casset et al., 2001); in consequence, H. curvispina acts as an omnivore species. Because H. pseudoazteca predates on the same algae that H. curvispina, the relationship between these two species is a mixture of predation and competition occurring simultaneously at different intensities. We ask ourselves if the system formed by the two Hyallela species is dynamically stable. To answer this question, it is necessary to study the behavior of the system under different conditions and to consider direct effects (predation and competition) and indirect ones (the refuge that macrophytes give to the prey amphipod). In this paper, we present a model for this system based on Lotka–Volterra type equations with combined interactions. This approach is the first to focus the problem of the combination of roles between competition and predation in the same species.
2. The model The biological system consists of three compartments (populations): algae (A), H. pseudoazteca (N1 ) and H. curvispina (N2 ). Biomass of each compartment has inputs and outputs that depend on both intrinsic and environmental variables. This system can be sketched as in Fig. 1. For H. pseudoazteca (N1 ) algae food and the refuge (given by macrophytes) are inflows, and mortality and predation are the outflows. For H. curvispina (N2 ) inputs come from predation on algae and amphipods, and the output is mortality. The model can be written as a system of ODE since it is proposed to represent the dynamics in a pool, a part of a stream. It is accepted that plain streams work as a sort of linked pools (backwaters) connected by rapids (Feijoo et al., 1999). So it can be considered that each pool has its proper populations, which have a local dynamics. The scheme shown in Fig. 1 can be expressed by a system of three coupled dimensionless equations of the form: dx = x(1 − x) dt
x M
− 1 − H1 xy − H2 xz
dy = −R1 y + e1 H1 xy − dt
dz = −R2 z + e2 H2 xz + e3 dt
KP2 ,
H3 =
KP3
(4)
where A = algae biomass (kg/m2 ), N1 = H. pseudoazteca biomass (kg/m2 ), N2 = H. curvispina biomass (kg/m2 ); e1 , e2 , and e3 are the parameters which corresponds to the assimilation efficiencies (0 < ei < 1), K is the carrying capacity of algae (kg/m2 ), is the growth rate of algae (day−1 ); P1 , P2 and P3 are predation efficiencies (day−1 kg−1 m−2 ); r1 and r2 are the growth rates of amphipods (day−1 ) which are negative in absence of preys (the signs are made explicit in the equations so that all the constants are positive, like in Lotka–Volterra type models). Parameter q denotes the number of refugee preys; m is the critical density of algae (threshold) needed to start to grow, according to previous models of periphytic and microbenthic algae dynamics (Momo, 1995; Saravia et al., 1999). M is the threshold expressed as a fraction of the carrying capacity of the algae K. Finally, t is the time but normalized to the intrinsic grow rate of the algae (t = T, T in units of days). The first term in Eq. (1) describes the monotonic changing of populations size with strong Allee effect for algae; this is a common observed phenomenon in microalgae (Momo, 1995; Saravia et al., 1999). The last two terms in Eq. (1) represent predation due to both species of Hyallela, with non-dimensional coefficients of predation H1 and H2 . H. pseudoazteca can be predated by H. curvispina but H. pseudoazteca can use macrophytes like refuge, showing a demographic barrier to predation because a fix stock Q of prey population individuals cannot be predated due to they are refugee. Macrophytes (refuge) have a dynamics several orders of magnitude slower than the dynamics of invertebrate and microalgae. In consequence, in periods of hydraulic stability (without floods), we can assume that macrophyte biomass remains roughly constant, therefore the number of refugee amphipods Q can also be considered constant. On the contrary, in that period (usually several months) amphipods and algae have an intense demography dynamics. The assumption Q = cte. is represented in the last terms of Eqs. (2) and (3). As a first approach to the model, we made several simplifications: the parameter H3 , the coefficient of predation of H. curvispina over H. pseudoazteca is considered a constant. Probably this assumption is not realistic because H3 should be a function of the density of algae (x); however, we made this approximation in order to simplify the dynamical analysis. The study of the model with non-constant H3 is the object of a future work. In the same way, we regard parameters e1 , e2 , e3 (conversion efficiencies) as constants since they are physiological traits of populations that can be considered as genetically determined; conversely, R1 and R2 are parameters that are strongly influenced by the habitat quality. Therefore, we consider these last two parameters to define the parameter space for our model.
(1)
H3 (y − Q )z
y≥Q
0
y
H2 =
H3 (y − Q )z
y≥Q
0
y
3. Results (2)
(3)
3.1. Analyzing the dynamics In order to make an outline of our problem, we obtain the fixed points of the system given in Eqs. (1)–(3). The fixed points are solutions of dx/dt = dy/dt = dz/dt = 0, that is the intersection of the
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nullclines of the system given by dx = 0 ⇔ x = 0 ∨ (1 − x) dt
x M
dy = 0 ⇔ −R1 y + e1 H1 xy − dt
− 1 − H1 y − H2 z = 0
H3 (y − Q )z
y≥Q
0
y
dz = 0 ⇔ z = 0 ∨ −R2 + e2 H2 x + e3 dt
(5)
=0
H3 (y − Q )
y≥Q
0
y
(6)
=0
(7)
The factor (y − Q) in the third term in Eqs. (6) and (7) represents the number of individuals y that is available to the predator z, in consequence it must be kept positive during the whole evolution. Therefore, when this term becomes negative, it is switched to zero (there are no available preys). From Eqs. (5) and (7) it can be seen that the yz and the xy planes are nullcline planes for x and z, respectively; as is the xz plane for y, when y < Q. From Eqs. (5)–(7), different equilibrium points Wi are obtained which represent different biological situations: W1 = (0, 0, 0). Extinction of the three species. W2 = (1, 0, 0). Only algae in its carrying capacity without amphipods. W3 = (M, 0, 0). Only algae in its threshold density. W4 = (x4 , y4 , 0) with 0 < x4 , y4 . Binary system without the omnivore amphipod. W5 = (x5 , 0, z5 ) with 0 < x5 , z5 . Binary system without the herbivore amphipod. W6 = (x6 , y6 , z6 ) with 0 < z6 < y6 ; 0 < x6 < y6 and 0 < Q < y6 . Tritrophic system; the herbivore amphipod is more abundant than the omnivore one. W7 = (x7 , y7 , z7 ) with 0 < Q < y7 , x7 < z7 and y7 < z7 . Tritrophic system; the omnivore amphipod is more abundant than the herbivore one. 3.2. Stability analysis and parameters The set of parameters in our model is divided in two groups. The first one constitutes the parameters that remain quite constant in the time scales considered. The assimilation efficiencies ei are strongly determined by physiology restrictions and genetics; the Pi ’s depend on morphologic features of the amphipods. Therefore, these parameters are included in the first subset. On the other hand, the mortality and growth rates have a strong variability in field due to their dependence on environmental conditions, hydrological conditions, etc. These parameters are included in the second subset. According to these considerations, we study numerically the dynamical response of the system defining our parameter space as a function of R1 and R2 . From Casset et al. (2001), we are aware that the mortality rates of amphipods are quite smaller than the growth rate of the algae. Therefore we propose values and ranges for the parameters as follows: Ri < 1,
Hi 1,
Q 1,
M 1,
ei 1
with
i = 1, 2
H3 < H1 , H2 , e3 < e1 , e2 e1 = 0.02,
e2 = 0.03,
e3 = 0.009,
H3 = 0.07,
H1 = 1.5,
H2 = 1.
M = 0.02,
Q = 0.01,
The values of the parameters were obtained from several field studies carried out in the Luján river basin (Buenos Aires, Argentina) and some laboratory experiments. Original data can be consulted in Giorgi and Malacalza (1994), Saravia et al. (1998, 1999), Giorgi et al. (1998) and Giorgi and Ferreyra (2000).
Fig. 2. Regions of stable solutions in the parameter space (R1 vs. R2 ). The different zones are discussed in the text. The values of the constant parameters are: e1 = 0.02, e2 = 0.03, e3 = 0.009, M = 0.02, Q = 0.01, H3 = 0.07, H1 = 1.5, H2 = 1. The numerical calculations are obtained for a grid of 40 × 40 values.
Defining our parameter space as R1 , R2 ∈ [0, 0.03], we can characterize the equilibrium points mentioned above, taking into account their biological meaning. Analyzing eigenvalues for the linearized system around a fixed point, we study the stability of the solutions Wi (Guckenheimer and Holmes, 1983; Momo and Capurro, 2006), finding stable and unstable solutions. The stable ones are presented in Fig. 2. W1 → stable solution in the part of the parameter space corresponding to the model: Regions I–II–III–IV (three real negative eigenvalues, atractor) W2 → unstable solution in the part of the parameter space corresponding to the model (two positive and one negative real eigenvalues, saddle node) W3 → unstable solution in the part of the parameter space corresponding to the model (one positive and two negative real eigenvalues, saddle node) W4 → stable solution in Region IV (three negative real eigenvalues, atractor) W5 → stable solution in Region III (three negative real eigenvalues, atractor) W6 → unstable solution in the part of the parameter space corresponding to the model W7 → stable solution in Regions II–III–IV (three negative real eigenvalues, atractor). Fig. 2 shows a summary of the cases previously discussed. The parameter space can be divided in several zones, where each one corresponds to one of the following coexistence of solutions: Zone I → extinction of three species (W1 ). Zone II → stable coexistence of the three species (W7 ) and extinction of three species (W1 ). Zone III → stable coexistence of algae and H. curvispina (W5 ), stable coexistence of algae, H. pseudoazteca and H. curvispina (W7 ) and extinction of three species (W1 ). Zone IV → stable coexistence of algae, H. pseudoazteca and H. curvispina (W7 ), stable coexistence of algae and H. pseudoazteca (W4 ) and extinction of three species (W1 ). 4. Discussion Throughout in all the parameter space it is possible to find a stable situation with the absence (extinction) of the three species. This is due by the strong Allee effect occurring in the algae growth. Effectively, algae biomass needs to reach a threshold (M) to have
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a positive growth rate and without algae, there is not support for amphipods populations. Another interesting behavior is found in zone II. In this region, the omnivore species has growth rates with high absolute values and, independently of the growth rate of the herbivore amphipod, there is the possibility of the stable coexistence of the three species. This is because the H. curvispina population decays faster than the H. pseudazteca one, avoiding the extinction of the prey amphipod. However, this occurs when the biomass of H. curvispina is higher than the biomass of the herbivore H. pseudoazteca, which means that H. curvispina is processing the highest fraction of the available energy and it is harming the other amphipod by competition as well as predation. This fact explains why in Pampean streams it is frequent to find higher biomass of H. curvispina whereas H. pseudoazteca is much less abundant (Casset et al., 2001). The coexistence of algae and only one of the amphipods species is restricted to zones III and IV. At zone III, when the growth rate of the herbivore is higher (in absolute value) than the omnivore one, the stable binary coexistence is between algae and the omnivore because the herbivore amphipod population decays at higher velocity than the other amphipod species. On the other hand, at zone IV, the reciprocal situation is stable (algae and herbivore) despite y > Q, so there are available preys for H. curvispina; the reason is again the difference between both net growth rate that, in this case, favors to the herbivore amphipod. It is important to mention that the behavior in each zone is highly dependent of the initial conditions. Further studies must be oriented to examine the size of the attractive basin for each equilibrium point or the effect of other population parameters, like efficiencies (Cassinari et al., 2007), that would determine chaotic behavior and bifurcation points. In our system, there is a strong bottom-up control given by the growth threshold of the algae. Moreover, we can consider that in the three species coexistence case, the ecological control can be characterized as top-down; this is only if we accept that the control can be characterized by the stocks of each species determining the amount of energy that flux throughout them. An exhaustive analysis of the energy fluxes would required another type of model since this one cannot allows us to analyze fluxes.
Acknowledgment This work was partially supported by CONICET (PIP 6124). References Casset, M., Momo, F., Giorgi, A., 2001. Dinámica poblacional de dos especies de anfípodos y su relación con la vegetación acuática en un microambiente de la cuenca del río Luján (Argentina). Ecología Austral 11, 79–85. Cassinari, M.P., Groppi, M., Tebaldi, C., 2007. Effects of predation efficiencies on the dynamics of a tritrophic food chain. Math. Biosci. Eng. 4, 431–456. Feijoo, C.S., Giorgi, A.D.N., García, M.E., Momo, F.R., 1999. Temporal and spatial variability in streams of a Pampean basin. Hydrobiologia 394, 41–52. Getz, W.M., 1999. Population and evolutionary dynamics of consumer–resource systems. In: McGlade, J. (Ed.), Advanced Ecological Theory. Principles and Applications. Blackwell Science, Oxford, pp. 194–231. Giorgi, A., Malacalza, L., 1994. Biomass variation of microphytobenthos in a plain stream. Verh. Internat. Verein. Limnol. 25, 1883–1887. ˜ P., Duttweiler, F., 1998. Annual variation of periphyGiorgi, A., Feijoó, C., Calvino, ton biomass in two plain streams with different macrophyte abundance. Verh. Internat. Verein. Limnol. 26, 1698–1701. Giorgi, A.D.N., Tiraboschi, B., 1999. Evaluación experimental del efecto de dos grupos de macroinvertebrados (anfípodos y gasterópodos) sobre algas epífitas. Ecología Austral 9 (1), 35–44. Giorgi, A., Ferreyra, G., 2000. Phytobenthos colonization in a lowland stream in Argentina. J. Freshwater Ecol. 15 (1), 39–46. González, E., 2003. The freshwater amphipods Hyalella Smith, 1874 in Chile (Crustacea: Amphipoda). Revista Chilena de Historia Natural 76, 623–637. González, E., Watling, L., 2003. A new species of Hyalella from Brazil (Crustacea: Amphipoda: Hyalellidae), with redescriptions of three other species in the genus. J. Nat. Hist. 37, 2045–2076. Guckenheimer, J., Holmes, P., 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York. Maionchi, D.O., dos Reis, S.F., Aguiar, M.A.M., 2006. Chaos and pattern formation in a spatial tritrophic food chain. Ecol. Model. 191 (2), 291–303. Momo, F.R., 1995. A new model for periphyton growth in running waters. Hydrobiologia 299, 215–218. Momo, F., Capurro, A., 2006. Ecología Matemática, Principios y Aplicaciones, Ediciones Cooperativas (and references therein). Naji, R.K., Balasim, A.T., 2007. Dynamical behavior of a three species food chain model with Beddington–DeAngelis functional response. Chaos Solitons Fract. 32 (5), 1853–1866. Pimm, S.L., 1999. The dynamics of the flows of matter and energy. In: McGlade, J. (Ed.), Advanced Ecological Theory. Principles and Applications. Blackwell Science, Oxford, pp. 172–193. Ramos-Jiliberto, R., Mena-Lorca, J., Flores, J.D., Morales-Álvarez, W., 2008. Role of inducible defenses in the stability of a tritrophic system. Ecol. Complex. 5 (2), 183–192. Saravia, L.A., Momo, F.R., Boffi Lissin, L.D., 1998. Modelling periphytons’s dynamics in ruinning waters. Ecol. Model. 114, 35–47. Saravia, Giorgi, Momo, 1999. A photographic method for periphyton chlorophyll-a estimation in artificial sustrata. Aquat. Ecol. 33, 325–330.