Structured population models are among the most widely used tools in ecology and evolution. Integral projection models (IPMs) use continuous representations of how survival, reproduction and growth change as functions of state variables such as size, requiring fewer parameters to be estimated than projection matrix models (PPMs). Yet, almost all published IPMs make an important assumption that size‐dependent growth transitions are or can be transformed to be normally distributed. In fact, many organisms exhibit highly skewed size transitions. Small individuals can grow more than they can shrink, and large individuals may often shrink more dramatically than they can grow. Yet, the implications of such skew for inference from IPMs has not been explored, nor have general methods been developed to incorporate skewed size transitions into IPMs, or deal with other aspects of real growth rates, including bounds on possible growth or shrinkage. Here, we develop a flexible approach to modelling skewed growth data using a modified beta regression model. We propose that sizes first be converted to a (0,1) interval by estimating size‐dependent minimum and maximum sizes through quantile regression. Transformed data can then be modelled using beta regression with widely available statistical tools. We demonstrate the utility of this approach using demographic data for a long‐lived plant, gorgonians and an epiphytic lichen. Specifically, we compare inferences of population parameters from discrete PPMs to those from IPMs that either assume normality or incorporate skew using beta regression or, alternatively, a skewed normal model. The beta and skewed normal distributions accurately capture the mean, variance and skew of real growth distributions. Incorporating skewed growth into IPMs decreases population growth and estimated life span relative to IPMs that assume normally distributed growth, and more closely approximate the parameters of PPMs that do not assume a particular growth distribution. A bounded distribution, such as the beta, also avoids the eviction problem caused by predicting some growth outside the modelled size range. Incorporating biologically relevant skew in growth data has important consequences for inference from IPMs. The approaches we outline here are flexible and easy to implement with existing statistical tools.
Structured demographic models are among the most common and useful tools in population biology. However, the introduction of integral projection models (IPMs) has caused a profound shift in the way many demographic models are conceptualized. Some researchers have argued that IPMs, by explicitly representing demographic processes as continuous functions of state variables such as size, are more statistically efficient, biologically realistic, and accurate than classic matrix projection models, calling into question the usefulness of the many studies based on matrix models. Here, we evaluate how IPMs and matrix models differ, as well as the extent to which these differences matter for estimation of key model outputs, including population growth rates, sensitivity patterns, and life spans. First, we detail the steps in constructing and using each type of model. Second, we present a review of published demographic models, concentrating on size‐based studies, which shows significant overlap in the way IPMs and matrix models are constructed and analyzed. Third, to assess the impact of various modeling decisions on demographic predictions, we ran a series of simulations based on size‐based demographic data sets for five biologically diverse species. We found little evidence that discrete vital rate estimation is less accurate than continuous functions across a wide range of sample sizes or size classes (equivalently bin numbers or mesh points). Most model outputs quickly converged with modest class numbers (≥10), regardless of most other modeling decisions. Another surprising result was that the most commonly used method to discretize growth rates for IPM analyses can introduce substantial error into model outputs. Finally, we show that empirical sample sizes generally matter more than modeling approach for the accuracy of demographic outputs. Based on these results, we provide specific recommendations to those constructing and evaluating structured population models. Both our literature review and simulations question the treatment of IPMs as a clearly distinct modeling approach or one that is inherently more accurate than classic matrix models. Importantly, this suggests that matrix models, representing the vast majority of past demographic analyses available for comparative and conservation work, continue to be useful and important sources of demographic information.
more » « less- Award ID(s):
- 1753954
- NSF-PAR ID:
- 10449093
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Ecological Monographs
- Volume:
- 91
- Issue:
- 2
- ISSN:
- 0012-9615
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
more » « less
Abstract Metapopulation models include spatial population dynamics such as dispersion and migration between subpopulations. Integral projection models (IPMs) can include demographic rates as a function of size. Traditionally, metapopulation models do not included detailed populaiton models such as IPMs. In some situations, both local population dynamics (e.g. size‐based survival) and spatial dynamics are important.
We present a Python package,
MetaIPM , which places IPMs into a metapopulation framework, and allow users to readily construct and apply these models that combine local population dynamics within a metapopulation framework.MetaIPM includes an IPM for each subpopulation that is connected to other subpopulations via a metapopulation movement model. These movements can include dispersion, migration or other patterns. The IPM can include for size‐specific demographic rates (e.g. survival, recruitment) as well as management actions, such as length‐based harvest (e.g. gear specific capture sizes, varying slot limits across political boundaries). The model also allows for changes in metapopulation connectivity between locations, such as a fish passage ladders to enhance movement or deterrents to reduce movement. Thus, resource managers can useMetaIPM to compare different management actions such as the harvest gear type (which can be length‐specific) and harvest locations.We demonstrate how
MetaIPM may be applied to inform managers seeking to limit the spread of an invasive species in a system with important metapopulation dynamics. Specifically, we compared removal lengths (all length fish versus longer fish only) for an invasive fish population in a fragmented, inland river system.MetaIPM allowed users to compare the importance of harvesting source populations away from the invasion front, as well as species at the invasion front. The model would also allow for future comparisons of different deterrent placement locations in the system.Moving beyond our example system, we describe how
MetaIPM can be applied to other species, systems and management approaches. TheMetaIPM packages includes Jupyter Notebooks documenting the package as well as a second set of JupyterNotebooks showing the application of the package to our example system. -
more » « less
Abstract A central debate in ecology has been the long‐running discussion on the role of apex predators in affecting the abundance and dynamics of their prey. In terrestrial systems, research has primarily relied on correlational approaches, due to the challenge of implementing robust experiments with replication and appropriate controls. A consequence of this is that we largely suffer from a lack of mechanistic understanding of the population dynamics of interacting species, which can be surprisingly complex. Mechanistic models offer an opportunity to examine the causes and consequences of some of this complexity. We present a bioenergetic mechanistic model of a tritrophic system where the primary vegetation resource follows a seasonal growth function, and the herbivore and carnivore species are modeled using two integral projection models (IPMs) with body mass as the phenotypic trait. Within each IPM, the demographic functions are structured according to bioenergetic principles, describing how animals acquire and transform resources into body mass, energy reserves, and breeding potential. We parameterize this model to reproduce the population dynamics of grass, elk, and wolves in northern Yellowstone National Park (USA) and investigate the impact of wolf reintroduction on the system. Our model generated predictions that closely matched the observed population sizes of elk and wolf in Yellowstone prior to and following wolf reintroduction. The introduction of wolves into our basal grass–elk bioenergetic model resulted in a population of 99 wolves and a reduction in elk numbers by 61% (from 14,948 to 5823) at equilibrium. In turn, vegetation biomass increased by approximately 25% in the growing season and more than threefold in the nongrowing season. The addition of wolves to the model caused the elk population to switch from being food‐limited to being predator‐limited and had a stabilizing effect on elk numbers across different years. Wolf predation also led to a shift in the phenotypic composition of the elk population via a small increase in elk average body mass. Our model represents a novel approach to the study of predator–prey interactions, and demonstrates that explicitly considering and linking bioenergetics, population demography and body mass phenotypes can provide novel insights into the mechanisms behind complex ecosystem processes.
-
more » « less
Abstract Integral projection models (IPMs) can estimate the population dynamics of species for which both discrete life stages and continuous variables influence demographic rates. Stochastic IPMs for imperiled species, in turn, can facilitate population viability analyses (PVAs) to guide conservation decision‐making. Biphasic amphibians are globally distributed, often highly imperiled, and ecologically well suited to the IPM approach. Herein, we present a stochastic size‐ and stage‐structured IPM for a biphasic amphibian, the U.S. federally threatened California tiger salamander (CTS) (
Ambystoma californiense ). This Bayesian model reveals that CTS population dynamics show greatest elasticity to changes in juvenile and metamorph growth and that populations are likely to experience rapid growth at low density. We integrated this IPM with climatic drivers of CTS demography to develop a PVA and examined CTS extinction risk under the primary threats of habitat loss and climate change. The PVA indicated that long‐term viability is possible with surprisingly high (20%–50%) terrestrial mortality but simultaneously identified likely minimum terrestrial buffer requirements of 600–1000 m while accounting for numerous parameter uncertainties through the Bayesian framework. These analyses underscore the value of stochastic and Bayesian IPMs for understanding both climate‐dependent taxa and those with cryptic life histories (e.g., biphasic amphibians) in service of ecological discovery and biodiversity conservation. In addition to providing guidance for CTS recovery, the contributed IPM and PVA supply a framework for applying these tools to investigations of ecologically similar species. -
more » « less
Abstract Structured compartmental models in mathematical biology track age classes, stage classes, or size classes of a population. Structured modeling becomes important when mechanistic formulations or intraspecific interactions are class‐dependent. The classic derivation of such models from partial differential equations produces time delays in the transition rates between classes. In particular, the transition from juvenile to adult has a delay equal to the maturation period of the organism. In the literature, many structured compartmental models, posed as ordinary differential equations, omit this delay. We reviewed occurrences of continuous‐time compartmental models for age‐ and stage‐structured populations in the recent literature (2000–2016) to discover which papers did so. About half of the 249 papers we reviewed used a maturation delay. Papers with ecological models were more likely to have the delay than papers with disease models, and mathematically focused papers were more likely to have the delay than biologically focused papers.
Recommendations for Resource Managers Interacting populations often are modeled with systems of ordinary differential equations in which the state variables are numbers of individuals of each species and interaction terms depend only on the current state of the system.
Single‐population continuous‐time models with age‐ or stage‐structure, in which state variables represent numbers of individuals in classes such as juveniles and adults, often but not always contain maturation time delays in the transition rates between classes. The exclusion of the delay typically changes the model dynamics.
Managers should be aware of the maturation delay issue when considering the results of continuous‐time models of structured populations.
Discrete‐time models have an inherent time delay, set by the census time step chosen by the modeler, and for that reason are convenient for modeling maturation and other biological delays.
