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Population Dynamics

DEB 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 'determines 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 would 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.

Ecosystems have

  • (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
Animals are consumers, so they feed on other organisms, sometimes even their own species, forming food-chains and food-webs. Parasites are a special type of consumers that are typically much smaller than their hosts and can effect their health. DEB models incorporate lethal effects via maturity maintenance, which interacts with the defense system and affects the hazard rate. Sublethal effects of parasites on their hosts are, like that of toxic compounds, incorporated via changes in parameter values. Parasite population dynamics has similarities with that of most viruses, which have a degenerated i-state. Epidemiology can be seen as a sub-field of population dynamics, with due attention for the health/illness dynamics of the (constant) host population. Not all plants, "algae" and bacteria are primary producers, while almost all of them also consume organic compounds to some extend (myxotrophy). Most fungi and bacteria specialize in the role of decomposers, frequently forming symbioses (e.g. mycorrhizas) with other organisms. DEB models for micro-organisms (bacteria, "algae") have a number of reserves that equals the number of possibly limiting substrates (nutrients, light). Plants additionally require multiple structures (root, shoot) and typically have more dynamic surface area to volume relationships, compared to animals. Stoichiometric constraints on growth and reproduction complicate these models somewhat, but so far these DEB models fit data very well. Since DEB models explicitly respect mass and energy conservation and specify product formation (faeces, N-waste, wood, metabolic water, etc), they can be used to model syntrophic relationships, symbioses, food-chain and food-web dynamics, nutrient recycling, etc. Since DEB theory formulates food acquisition in the framework of Synthesizing Units (which is Markovian stochastic by nature), and the ageing process on the basis of hazard rates, while parameter values are individual-specific, DEB-based population dynamics can naturally include stochasticity and has the ideal structure for Adaptive Dynamics (AD), apart from the problem that AD rapidly becomes complex in the case of coupled traits.

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, while DEB theory actually implies that small-bodied individuals follow fluctuations in food density faster than big-bodied ones. 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, although event-driven stochastic ode's have been formulated that respect mass conservation. 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 an associated 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. The crudest of all approaches assumes that this still applies when food density is (slowly) changing in time and reduces population dynamics to a macrochemical reaction equation with time-varying coefficients. 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.
The strength of DEB models is that the most detailed method (ibm's) can be used for providing a reference, to study how much realism is lost by less detailed methods, before further steps towards ecosystem dynamics are made. Also the crudest approach (pss's) is straightforward to implement. The methods between these extremes are much more difficult, but comparison between methods might help judging to what extend "details" matter. Future developments might include hybrid methods, predictor-corrector style, that specify criteria for p-states, to identify moments when a less-detailed method should temporarily be replaced by a more detailed method and vice versa, which might be species-specific. The general idea is to implement as less detail as possible stepping up from individuals, via populations, to ecosystems, without losing too much realism.

From AmP entries to population dynamics

DEBtool, 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
  1. 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.
  2. 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: stage-specific, but 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, since the implied increase in feeding by cohort members is compensated by a decrease in numbers. 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. Other species, with less offspring don't survive thinning in the long run.
    Since these causes are independent, the total hazard rate equals the sum of the cause-specific hazard rates
  3. specification of the founding population (p-state at time zero)
  4. prescribed trajectories for temperature and functional response or extra parameters for the reactor settings
For the time being, realistic topics, such as i-state dependent food selection (as specified by DEB theory), are ignored and we work with a single food type. It makes little sense to think about these phenomena without a proper understanding of simpler cases.

To accomodate the energy cost of male production in the case of gonochoric species in a DEB-context, reproduction efficiency is halved on the assumption that the sex ratio is 1:1, since any difference between sexes typically develop slowly during ontogeny. Many entries have parameters for males that deviate from that of females, which and be taken into account in a straightforeward way.

Pseudo steady states (pss's) in AmP

All entries have a page "Population traits", where the pss-approach has been worked out in presence and absence of thinning. Apart from a very high food level, a low one for which the population growth rate is zero is shown, as well as an intermediatiary scaled functional response, in the case where males and females have the same parameters. If the parameters differ between sexes, the latter is suppressed to avoid too large tables. If a table has NaN's (Not-a-Number), this typically means that the species does not survive thinning. The scaled functional response for which the specific population growth rate is zero relates to the female parameters; males might not reach puberty at this food level, with NaN's as result. Non-smooth looking survivor curves also originate from numerical problems (and are not model implications), typically caused by puberty being reached only very close to the ultimate size.

The fact that most species reproduce in clutches (or litters), typically in response to a cyclic behaving environment, rather than one-by-one, is difficult to accomodate in the pss-approach, which assumes that the environment stays constant (in terms of temperature and food). Clutches reduce the population growth rate, due to the interest-upon-interest principle, where the very first young contributes most since it is first ready to reproduce itself, while clutches require longer accumulation time of the reproduction buffer.

The species-subdirectories of the directory entries_web, where the population-trait-pages are parked, also have a file popStat.mat, i.e. a structure that has all numbers shown on these pages. The fuction write_popStat collects these structures, via the web, into one for all species, similar to allStat.mat. This can be handy for comparison of population traits amoung species.

The population-trait-page can also be generated in the context of paramater estimation, by choosing estim_options('results_output', 6) in run_my_pet.m, i.e. one of the 4 files that are required to estimate parameters from data in the AmP-system. The pages have a default-setting of background hazards equal to zero. Independent from the estimation context, the user can choose different values for background hazards with AmPtool function prt_my_pet_pop, which also holds for other values to temperature and scaled functional response. In that case the parameters are extracted from allStat.mat. If hazards are chosen too high, long-term survival is not possible and functions even might become holted.