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Population DynamicsDEB theory has been set up for the purpose to specify (and understand) rules for how individuals interact with the environment that, in combination with models for environmental dynamics, determine population and ecosystem dynamics. Models for environmental dynamics should specify substrate dynamics (including transport and geochemical transformations), and the (micro)-climate (temperature and, in terrestrial environments, the dynamics of water). Given the DEB and environment models, population dynamics can be seen as an advanced book-keeping exercise which affects survival probabilities; ageing is rarely the cause of death of small-bodied species in natural populations, but still is important in population models to guarantee that individuals do not stay forever in the population. It sets a maximum to the memory of populations: population dynamics not only depends on the present state of the population (and environment), but also on its state in the (recent) past. This latter is usually taken into account by extending the states of individuals (e.g. including the ageing process and reserve). The book-keeping exercise is complex and, in practice, one is forced to apply a number of simplifications that raise problems by themselves. To structure the book-keeping, the (vector-valued) state of individuals is called the i-state (e.g. age, maturity, reserve, structure, reproduction buffer), and that of the population the p-state, i.e. the numbers of individuals with a given i-state for all possible values of i-states.
Population dynamics is typically seen in the light of the ultimate aim of ecology: the understanding of ecosystem dynamics. Ecosystems are a (large) set of interacting populations, the most important interaction being nutrient recycling. Predation-prey dynamics attracted a lot of attention in the literature, but its ecological significance in this context is limited by the difficulty of selecting just two players from a complex interplay: if predator dynamics depends on prey availability, why does this not apply to prey dynamics? These models typically just count numbers of individuals in a population, ignoring properties of individuals, which classify them as unstructured population models, to differentiate them from (physiologically) structured population models. Models that assume homogeneous (well-mixed) space are called spatially implicit, to differentiate them from spatially explicit models. Spatially implicit models are typically used to analyse model properties, not to produce realistic predictions. The analysis can focus on transient behaviour, i.e. population-trajectories (functions of the p-state as functions of time), or asympotic (i.e. very long term) behaviour. The study of asympotic behaviour as function of parameter values is called bifurcation analysis.
- (primary) producers, which extract energy from light (plants, "algae", blue-green and sulphur bacteria) and/or minerals (some bacteria)
- consumers, which feed on primary producers, decomposers and consumers
- decomposers, which mineralize organic compounds (produced by primary producers and consumers) and close mineral recycling
The following approaches, ordered from detailed to crude, can be used to follow the dynamics of populations, each with their own strong and weak properties.
- Individual-based models (ibm's): This approach follows each individual through time, possibly including spatial coordinates in inhomogeneous space. If space is discretized, and migration rules are specified, the models classify as cellular automata. The concept "individual" has been extended to "agent" to include social behaviour, leading to the term agent-based models (abm's). Spatial structure and interactions between individuals pose no problem for ibm's, although this can rapidly increase computational requirements. Ibm's make use of the discrete nature of individuals, but a large number of individuals is a problem for this approach. For this reason, ibm's are rarely used for micro-organisms or other small organisms that can build up huge numbers of individuals.
- Partial differential equations (pde's): Individuals change their i-state continuously; pde's follow the changes in the number of individuals with similar i-states. The most popular methods to integrate pde's numerically to arrive at population-trajectories, discretize the values of (continuous) i-states and follow cohorts of similar individuals through time. The pde approach has been developed further to a coupled system of nonlinear renewal equations and differential delay equations for the analysis of asymptotic behaviour of pde's. Spatial structure can be incorporated, using Lagrange-methods, but rapidly increases computational requirements. The number of individuals can be large without imposing problems for pde's, but the discrete nature of individuals does. All individuals potentially contribute to the population-reproduction process. This is usually not a problem, but if food availability is very poor, such that no individual can produce a single offspring, the sum of such individuals still can in the pde-approach (which is unrealistic). Direct interactions between individuals pose problems and can only be implemented via interaction with the environment. The dynamics of structured populations feeding on other structured populations is really complex. Considerable work has been invested for how to combine pde's for the small-bodied (numerous) individuals with ibm's for the larger-bodied (less numerous) ones in multi-population models.
- Matrix models (mm's): The population is sub-divided in a limited number of classes corresponding to the states of individuals, similar to pde's, and also time is dicretised. A state vector counts the number of individuals in each class, and a matrix of transition probabilities specifies possible changes in the state of individuals. The set-up is very similar to discrete-time, discrete-state Markov-chain, the continuous-time version is called Markov-process. The mm-approach has become popular in ecology, since it is computationally simple and fast. Where pde's still follow individuals, mm's only deal with changes in states, not how long an individual has been in that state, for instance. Integral projection models (ipm's) avoid discrete state-classes, linking to discrete-time, continuous-state Markov-chains. This can be implemented with standard matrix software. Both mm's and ipm's classify as demographic models, since changes in i-state depend on time only, not on i-state. This can present a serious drawback for DEB applications. State-structured models always reduce to age-structured models in constant environments. If food densities fluctuate, however, this is no longer the case and this mismatch can have strong effects: mm-generated population-trajectories can deviate substantially from the ones implied by DEB models, due to the reduced book-keeping. This reduced book-keeping also complicates explicit mass and energy conservation, which makes it hard to model nutrient recycling.
- Ordinary differential equations (ode's): Depending on the model for individuals, pde's for populations can sometimes be reduced to a finite set of ode's that include particular changes in i-states, sharing the advantages of pde's over ode's. This is called the linear chain trick, where not only numbers of individuals, but also total length (of all individuals combined), surface area and volumes are followed. It can only be applied to DEB models using additional approximations, such as assuming equal reserve density for all individuals. The simplification is both the strength and the weakness of ode's, and still attractive if changes in states of individuals have little effect on population dynamics. While for micro-organisms this might hold by rough first approximation, for other organisms this might not be realistic: the contribution of neonates to reproduction easily dominates the increase in numbers because of the interest-on-interest principle. Energy and mass conservation can only be respected in a very crude way with the ode-approach. Population models in the past (before 1980 say) ignore the state of individuals and just follow total mass or number of individuals. Since pde's can been seen as an infinitely large set of coupled ode's, it is easy to see the substantial simplification that is obtained by ode's, compared to pde's. This simplicity allows the partitionning of space into cells, each with its own population and immigration/emigration rules: the meta-population models.
- Pseudo steady states (pss's): Given a constant food level, populations eventially grow exponentially and (generally) have a linked stable size (and age) distribution. The rate of population growth can be found by solving the characteristic equation. DEB models specify all mass that is consumed (food, dioxygen) and produced (feaces, minerals, dead bodies) by the population. This allows the crudest of all approaches by assuming that this still applies when food density is (slowly) changing in time and reduce population dynamics to a dynamic macrochemical reaction equation. The method can be close to accurate provided that changes in food density are slow enough (in relation to growth and reproduction) and can be valuable in the context of biotechnology and ecosystem dynamics.
From AmP entries to population dynamicsDEBtool, in combination with AmP results, focuses on the first steps in the direction of ecosystem dynamics: population dynamics in a generalized well-stirred reactor, and allow the comparison of several methods the follow population dynamics. Such a reactor has a substrate (nutrients, light, food) input, and an output of substrates, individuals and their products, each at a controlled rate. The task of population dynamics is here to specify the dynamics of the contents of the reactor, respecting mass and energy conservation. Fed-batch reactors are a special case of generalized reactors: they have no output. The step from AmP entries to population dynamics needs the following
- extra parameter(s) for the specification of the reproduction buffer handling rules (spawning), respecting the ecotype for reproduction, gender and possibly migration and embryo environment. DEB theory has a number of frequently occuring rules.
- extra parameters for the hazard rate given the i-state.
Basic causes of death are
- aging: part of all AmP entries
- starvation: part of DEB theory, but not applied in AmP
- accidental: independent of the i-state
- thinning: not part of DEB theory. If the hazard rate equals r×2/3 for isomorphs, or r for V1-morphs, where r is the specific growth rate of the structure of a DEB-individual, the total feeding rate of a cohort of individuals does not change during growth at constant food. This can be used to avoid that a very large number of tiny neonates from a single mother turns population dynamics unrealistic if they survive too long in the model
- specification of the founding population (p-state at time zero)
- prescribed trajectories for temperature and functional response or extra parameters for the reactor settings
Most ready-to-use software can not accomodate males in the case of gonochoric species. To approximate population dynamics of gonochoric species with this software, assuming that the sex-ratio is 1:1 and parameter values for males are close to those of females, females reproduction efficiency is halved, and the numbers of individuals is doubled.