Models of patch dynamics in tropical forests

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Models of Patch Dynamicsin Tropical Recent interest in the ecology and evolution of metapopulations and conservation of fragmented populations has stimulated the development of models that combine pakk and population dynamics in tropical forests. One approach uses matrix models that are actual metapopulation or multiregional demographic mode/s. Another approach uses computer simulations to model forest successionbased on the 6ehavior of individual trees. We review applications of 60th types of models and suggest new combined modelling approaches.

In 1970 Levin defined metapopulations as groups of local populations and proposed the first model of metapopulation dynamics’r2. As natural habitats become fragmented by human perturbation, and ecologists and evolutionary biologists become interested in understanding the processes and mechanisms underlying the observed patterns of populations and communities, the study of recently metapopulations has reawakened. Theoretical population ecologists2, population geneticists3 and conservation biologists2 metahave proposed new population models and developed field methods to generate the appropriate parameters. Tropical biologists have recently started to develop these models4-9, which hold great promise for basic and applied research in rain forests. In tropical rain forests, metapopulation models should consider a population growing in a dynamic environmental mosaic whose structure changes with time. The mosaic is created by the recurrent treefalls that open canopy gapslo. Hence, populations are spatially segregated in patches of vegetation at different successional stages”. Recent gaps have different light, nutrient, soil, humidity and vegetation conditions compared to closed-canopy sites’*-+. Gaps vary in size and shape, but can be defined as an opening in the canopy extending to within two meters from the groundI (but see Refs l&19). Elena Alvarez-Buylla is at the Centro de Ecologia, UNAM, Aptado Postal 70-275, Deleg. Coyoadn, DF 04510, Mexico; Rairl Garcia-Barrios is at the Centro de Investigacidn y Docencia Econ6micas, Aptado Postal 10-883, Carretera MCxico-Toluca Km 16.5, Cal. Lomas de Sta. Fe, Deleg. Alvaro Obreg6n, DF 01210, MCxico. Q 1993, Elsevier Science Publishers

Ltd (UK)

Forests ElenaR.Alvarez-Buyllaand RatitGarcia-Barrios Gaps close by growth of surrounding vegetation into the gap and by germination and growth of seedlings and saplings in the gap itself. The importance of patch dynamics in species population dynamics and life history evolution has been amply discussed for tropical rain forests20-25, but formal mathematical tools have been applied only recently. Two main modelling approaches have been used. The first one uses matrix models to describe population dynamics in each patch type and Markov chains to model the dynamics of the forest mosaic. Such models are therefore actual metapopulation or multiregional demographic models, and generally they incorporate linear behaviors only and may be solved analytically. They have been applied to particular plant species at the population level. The second approach uses computer simulations to model forest succession based on the behavior of individual trees. These models incorporate many species, and more biological detail, than the first type. They are never solved analytically and are not really metapopulation models because they do not consider the dynamics of populations in different patch types of an explicitly defined mosaic. Instead, they model succession in forest plots of even disturbance age. Matrix models

To model metapopulation dynamics in rainforest mosaics, one should consider the demographic behavior and dynamics of populations in each patch type, the transitions among types of vegetation patches through gap formation, succession and gap closure, and the movement of individuals among different patch types. In addition, these three submodels should be coupled. The simplest way to couple the population dynamics to the dynamics of the forest mosaic is by assuming that the dynamics of vegetation patches is independent of the population

dynamics of any particular species. This is a valid assumption for species-rich communities such as most tropical rain forests. &tch-specific population dynamics Standard matrix models have been used to analyse population dynamics in each patch type. These models have been recently reviewed26s27 (see Box 1). The demographic parameters in each patch type may be simulated according to theoretical propositions or may be estimated in the field. Horvitz and For example, Schemske5 used three models of increased demographic cost at all life stages of the perennial herb Calathea ovandensis (Marantaceae), as a function of patch age (linear, slow nonlinear, and fast nonlinear). Other studies have obtained field estimates6-9, which are important because the demographic response of some species to succession may be complex and vat-y among different life-stages.

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pbtch or successionaldynamics All authors have assumed a finite linear Markovian process to model the forest mosaic dynamics. A finite Markovian process is a stochastic process for which there is a finite number of possible states (patch types or successional stages) and in which the transitions of one state to another are fixed conditional probabilities28 (transitions among patch types). These are defined as the probability that state k occurs given that state I occurred in the previous period. In addition, one needs to know the initial probabilities with which each state occurs (a vector with the proportion of patches or area per type at time zero). We represent this process with a transition matrix, D, whose entries, dk/, represent the constant probabilities of a type I patch becoming a type k patch from t to 1 + 1. Defining a vector, f, with the number of patches or area per type, changes in the forest mosaic are modelled as: f(t+l)=Dxf(t)

(2)

The dominant eigenvalue of D equals one and its corresponding right eigenvector contains the proportion of patches or area per patch type in stable conditions. Patches in a forest mosaic can be classified according to different criteria. The number of patch types yields the dimension of matrix D. Published patch models for tropical rain forests have followed one of two alternatives, depending on whether they assume equal patch size or not. Horvitz and Schemske5 assumed all treefall gaps having the same size, and distinguished patches according to age only, with f containing the number of patches per type. They assumed a logistic relationship between the probability of gap formation and the age of the patch. Gaps in rain forests are not all of equal size, however, and their formation and closure rates vary with size29. Furthermore, gap size has significant effects on species’ demography30. Hence, models allowing for the formation of gaps of different sizes are necessary. For parameters, these models generally use data from periodic censuses of mapped portions of the forest mosaic3’ and the vector, 202

f, contains the areas per patch type rather than number of patches. Martinez-Ramos et a1.6 distinguished closed sites from recent gaps of different sizes; AlvarezBuylla and collaborators8,9 distinguished four patch types according to age and size. Metapopulation dynamics To model overall population dynamics, the patch-specific population model is coupled to the Markov chain model of forest dynamics. All possible transitions among life stages and patch types are contained in a matrix, G, with a dimension equal to the number of stage classes (s) times the number of patch types (n) (e.g. 10 x 10 if 2 patch types and 5 life stages are distinguished), and comprised of rr2 (four in this example - all the possible transitions among patch types) s x s submatrices (5 x 5 in this example). The metapopulation dynamics is then modelled by: n(t+

l)=Gxn(t)

(3) Where n (t) and n (t + 1) represent (s 5 n) 5 1 population vectors in successive times containing the number of individuals (nil) per stage category, i, and per patch type, 1, and the entries of G, g/i are the transition probabilities from stage j to stage i and from patch type 1 to patch k. If spatial heterogeneity without patch or successional dynamics is considered, the metapopulation matrix would have the patch-specific population submatrices along the diagonal with all other values equal to zero. A matrix incorporating patch dynamics would have entries in all those submatrices for which valid transitions among patch types were found according to the Markov matrix D. The probabilities g? are estimated by multiplying the transition probability from stage j to stage i by the transition from a type 1 patch to a type k patch because these two transitions are assumed to be independent from each other (see above). In principle, either the demographic transition in patch 1 or that in patch ,k could be used to compute the g/j .If the transition among patch types occurs right after the population census, then the demographic

transition considered is that in patch type 1; if it occurs before the census, then the demographic transition considered is that in patch type k. Dispersal above metapopulation The model does not provide an explicit formalization of the movement of individuals among patch types due to active movement (e.g. animals) or seed dispersal in plants. All published metapopulation models for tropical forests are for plants. In these, seed production and dispersal away and into different patch types change before the forest mosaic attains stable conditions. Horvitz and Schemske5 simulated dispersal in matrix G by subtracting and adding the fraction of seeds dispersed away and into a patch type, respectively, depending on the relative frequency of each patch. Seed production and dispersal as a function of the forest mosaic structure has been explicitly formalized in models applied to a tropical pioneer tree species Cecropia obtusifolia (Moraceae), at Los Tuxtlas, Mexico819. Applications of matrix models

Patch dynamics models in tropical rain forests as described above have been used to address questions concerning the dynamics, regulation and structure of plant populations 5-9, the evolution of life history traits of plant species in heterogeneous environments4-9, and the management of forests8f9. Martinez-Ramos et aL6 simulated the effect of changes in the frequency and size of gaps on the metapopulation growth rate (h) of the understory palm Astrocaryum mexicanum in a forest with two types of patches. They found that populations of this species may be at equilibrium or increasing under a wide range of forest gap regimes depending on gap size. Alvarez-Buyha’ quantified the combined and isolated effects of density dependence and patch dynamics by applying four matrix models to data from C. obtusifofia and mapping the contrasting results of population growth rate, size and structure to the models’ assumptions9. Density-dependent mechanisms in fecundity and adult

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survival seem to regulate the population and affect its structure. The size and frequency of gaps affect population size and growth rate. This study also illustrated that, in general, the effect of density dependence may be misvalued if patch dynamics is not considered. Alvarez-Buylla and Garcia-Barrios proposed a metapopulation model to study the relative contribution of the seed rain and the seed-bank in gap regeneration of pioneer tree specie@. Assuming a constant seedrain, an equilibrium analytical solution was obtained for the number of soil seeds per patch type. General behaviours of the seedbank depending on rates of soil seed survival, seed-rain and forest disturbance were discussed. For C. obtusifolia at the Los Tuxtlas, this model predicted equilibrium seed densities per patch type very similar to independent field estimates, and estimated that more than 90% of the seedlings established per year in a 5 ha plot come from seeds less than one year old Metapopulation models may be used also to quantify the demographic consequences of life history variation in patchy environments. 3Lmay be interpreted as the average overall fitness in longlived organisms with overlapping generations32. If conditions are assumed to be constant, the sensitivity coefficients of 3Lfrom a matrix model may be interpreted in terms of the evolutionary potential of genetic variations that could appear at different levels of the life cycle of a species27. Horvitz and Schemske5 examined the effects of seed dispersal on mean population fitness in Calathea ovandensis, which is dispersed by ants. They used numerical simulations to evaluate the effect on h of within-patch dispersal, long-distance dispersal and seed dormancy. They showed that slight changes in the demographic cost of forest succession or in the gap formation rate significantly modified the effects that different modes of dispersal had on h. They predicted strong selection for local dispersal to safe sites, weak selection for seed dormancy and selection against long-distance dispersal. These results are consistent

with field evidence of the life history of C. ovandensis. Alvarez-Buyha showed that the responses of h to possible selection pressures depend on whether or not patch dynamics is incorporated into the model. Metapopulation models may also be useful to address management issues. Alvarez-Buylla and Garcia-Barrios8 found that increased disturbance rates affect seed availability and mosaic structure less if patches 2-35 years old are opened than if mature patches I> 35 years old) are opened. They also found that extensive deforestation could prevent regeneration of pioneer species that, like Cecropia obtusifolia, have low soil seed survival rates. The effect of different harvesting regimes of adult trees has also been tested9.

In FORET, the vertical structure of the canopy is modeled explicitly and the light environment for each individual is defined by constructing a vertical profile of leaf area and available light based on the sizes of individuals. Individual growth is calculated using a species-specific function that predicts an expected growth increment given a tree’s current dimensions under optimal conditions. Growth is restricted depending on the vertical profile of light availability in the plot and other resources. The death of individuals is stochastic, and includes both age-related and stressinduced mortality. Disturbances are introduced as causes of increased individual mortality. Establishment and regeneration are stochastic, with maximum potential rates constrained by the same environmental factors that modify tree growth33-35. Species are generally grouped in a few discrete categories to model their response during establishment, growth and mortality36. Tests and applications of gap models focus on the average behavior of a set of plots. Typical results of forest simulators are successional sequences in an average plot along approximately 500 years34s35. Simulated data of species composition and abundance at different years are compared to observed data for stands of particular ages.

Simulation forest models Matrix population models do not incorporate explicitly the environmental and physiological factors that affect individuals. In contrast, individual-based forest simulators consider the life history of each individual in a community, incorporate modifications of the environment by the individuals and accidents of mortality and establishment that are amplified through nonlinear processes of plant-environment feedbacksY3. In these models the successional dynamics of forest stands is then simulated by integrating the behaviors of individuals33. Although they are not strictly metapopulation models, they can be used to evaluate the role of Applications of forest simulators disturbance on populations and The FORICO model is a tropical communities by incorporating in- forest succession model that was creased individual mortality rates. developed to investigate the ecoTwo simulation models have logical role of major and minor disbeen developed for tropical rain turbances on a montane rain forest forests: FORKO for a Montane rain in Puerto Rico, where tropical forest in Luquillo, Puerto Rico34, storms are prevalent34. Thirty-six and KIAMBRAM for an Australian sub- species with their autoecological tropical rainforest near Queensparameters were used. The densityland35. Both are of the FORET family diameter distributions of simuof the so-called ‘gap models’ that lated and observed patterns were simulate the life cycle of trees on very similar. Simulations including an annual time-step and on a plot hurricanes (i.e. with higher treefalls of defined size that corresponds than those produced by normal approximately to the zone of in- tree death) yielded forests with fluence of a single individual of dominance-diversity curves, species maximum size (c. 0. I ha). Each in- rankings and diversity figures that dividual on the plot influences, were more similar to those found and is influenced by, the growth of in the tabonuco forest than those all others, assuming horizontal ho- from simulations that did not inmogeneity inside the plot33t35. clude hurricanes. In the absence of 203

revi,ews

hurricanes, the model predicted a lower number of overall species and a forest dominated by latesuccessional and primary species. Various disturbance rates were tested and higher diversity was found for intermediate levels of disturbance. The KIAMBRAM mode1 was devised to mode1 forest dynamics, the role of treefalls and harvesting in an Australian subtropical rainforest near Queens1and35. It simulates the dynamics of 125 species. The model was verified by comparing the simulation results with composition of stands of known age in Lamington National Park, Queensland and validated by predicting the abundances and basal areas of trees in mature forests at Wiangaree State Forest (New South Wales). The model has been used to evaluate the effect of different timber harvest schemes for Australian subtropical rain forest. Conclusions and Perspectives Available applications of metapopulation models have provided general results concerning population dynamics, evolution and management of plant species in forest mosaics and have been successfully applied to particular species with a phenomenological perspective. Forest simulators, on the other hand, have the advantage of incorporating many species and more biological detail of individuals’ behaviour, but at the expense of analytical solutions and the capacity of obtaining genera1 conclusions. These simulators are tailored for particular forests and have been quite successful in practical studies on a small spatial scale33. One way to link the population matrix models to the many species simulation models would be to develop the Markovian analytical models of forest succession based on a plant-by-plant replacement process. This approach first formalized by Horn 37 has been developed and applied to tropical rain forests by Acevedo3a and Hubbell and Foster39, but these applications have not considered explicitly the dynamics of the forest mosaic. Patch dynamics could be incorporated by making transitions among species grouped in ecological guilds, depend on patch 204

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type and by explicitly considering transitions among patch types. For example, changes in the forest mosaic caused by tree-falls of species of particular life histories could be coupled to the dynamics of species with contrasting life histories. Such models should be useful to explore the mechanisms and conditions of coexistence of species with contrasting life histories in tropical forests, under various temporal and spatial regimes of forest disturbance. Nonlinearities could be introduced in these models, although in this case, analyses would require a simulation approach9v38. The same will be true for models that explicitly consider the spatial patterns of both plants and disturbances in a forest, that are both likely to affect population and community structure and dynamics. Finally, consequences of patch dynamics on the genetic composition and evolution of species should be addressed in future models?. Acknowledgements This research was financed by a scholarship from Universidad National Autdnoma de Mexico (Mexico) and a ‘Dora Garibaldi’ scholarship from the University of California at Berkeley to EAB, an improvement dissertation grant from NSF (BSR88-15681) to EAB, a Sigma Xi research grant and a dissertation research grant from UC-MEXUS to EAB. We acknowledge M. Martinez-Ramos and M. Slatkin for useful discussions and 1. Meave for a careful revision of a previous draft. Two anonymous reviewers made useful comments to improve previous versions.

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