Presented By: Dissertation Defense - Department of Mathematics
Dissertation Defense: Coexistence through life-history variation revisited in tractable models with explicit patch aging and/or size-structure
Ursula Trigos-Raczkowski
Abstract: A persistent puzzle in ecology is understanding the mechanisms of competitive coexistence in nature. One of the major hypotheses of how competitive coexistence can arise, in a variety of systems, but especially among tree species in forests (which are important given their role in the global carbon cycle), is that species differ in their successional niches. However, much work remains to fully understand the differences in demographic strategy that could mediate this successional niche differentiation, and the degree to which it is at play in nature. Mathematical modeling has proved to be an important tool for quantifying the influence of differential demographic strategies. However, modeling successional niche differentiation requires consideration of both patch-age structure and size-structure within competing populations. These have been only sparingly included in mathematical models due to the tendency to make the models too complicated to study analytically. In this thesis, we develop a suite of mathematical models that include these structures and systematically assess—using analytical and numerical approaches—their potential to give rise to opportunities for variation between species among demographic trade-offs to enable coexistence.
The first set of models we consider incorporate patch-age structure—defined as the subsequent progression of competitive dynamics, or the ‘aging’ of patches after disturbance events. Patch-age has been posited as creating opportunities for coexistence of species competing on a landscape if they differ in their life history (demographic) strategies. Mathematical models used to study this possibility are either too simple, not modelling patch aging explicitly, or far more complex and requiring extensive simulation. We study four patch-age structured nonlinear partial differential equation (PDE) models that are still analytically tractable in most cases but allow explicit consideration of the progression of competitive dynamics as patches age after disturbance. We consider three possible types of density-dependence for their importance in coexistence under disturbance: 1) on reproduction, 2) on recruitment, or 3) on mortality. Although in nature all three types of density-dependence are likely acting in concert, here we consider each in turn on its own, both to help retain analytical tractability, and to identify which type of density-dependence is critical to set up conditions for coexistence.
Under density-dependence on recruitment alone, the model does not permit feasible coexistence. However, under density-dependent reproduction or under density-dependent mortality, variation between two species along a reproduction survival and a sensitivity-to- competition/mortality trade-off both allow for feasible and stable coexistence. One species must have both higher reproduction and higher mortality than the other species in order for coexistence to occur. Inter mediate relative reproduction and death rates, which could arise from intermediate values of disturbance, lead to a wider coexistence region. The results we present contrast importantly with the classical competition-colonization trade-off model by showing that simply a reproduction/survival trade-off can facilitate coexistence under disturbance. This trade-off is arguably more often observed in nature.
The second set of models we consider incorporate size-structure. Models capturing the size-structured nature of competition between trees are complex, and have faced barriers to complete analysis, leaving the specific demographic trade-offs that could allow for coexistence unclear. We study four size-structured nonlinear PDE models which include hierarchical size-structuring while remaining analytically tractable. We again consider three possible types of density-dependence 1) on reproduction, 2) on recruitment or 3) on mortality.
Under only density-dependent mortality we find numerical analysis is needed. While we find that coexistence is not possible under density-dependent recruitment alone, we are able to show that hierarchical size-structuring allows for stable coexistence under density-dependent reproduction when species vary along a reproduction and maximal-height continuum. One species must have a higher reproduction rate while the other is able to grow much taller and escape the harmful effects of competition in order for coexistence to occur. This result provides a more rigorous demonstration of prior results from a more complex model which was analyzed numerically, and also highlights that a trade-off that has been observed empirically among tree species can result in coexistence.
The third set of models we consider employ both size and patch-age structure simultaneously. Size and patch-age structures have been used together through extensive simulation and a variety of trade-offs were found to influence coexistence. We consider a size and patch-age structured model which has both a size hierarchy as well as an explicit patch-age structure. We consider a density-dependence on 1) reproduction or 2) recruitment. Under density-dependent recruitment the model does not permit feasible coexistence. However, under density-dependent reproduction—and when shading effects are felt from all individuals—we show coexistence can occur through a trade-off between growth and survival. (Analysis of a density-dependent reproduction model where shading occurs only from taller individuals is in progress.) The growth/survival trade-off has been widely observed among tree species. The coexistence enabling nature of variation along this trade-off contrasts with forest ecologist’s historical focus on a trade-off between growth in high light and survival in the shade as the key to coexistence.
The results we present in this dissertation are the first (without assuming perfect plasticity or discrete size classes) to show analytical trade-offs resulting in coexistence through models which include hierarchical size-structure, patch-age structure or size and patch-age structure. Overall this dissertation provides a basis for an analytically tractable modeling approach for studying competition between two species (with the potential of extending to multiple species) whose populations are structured by size and/or patch-age and hence provides the core methodology needed for studying coexistence through life-history variation analytically.
The first set of models we consider incorporate patch-age structure—defined as the subsequent progression of competitive dynamics, or the ‘aging’ of patches after disturbance events. Patch-age has been posited as creating opportunities for coexistence of species competing on a landscape if they differ in their life history (demographic) strategies. Mathematical models used to study this possibility are either too simple, not modelling patch aging explicitly, or far more complex and requiring extensive simulation. We study four patch-age structured nonlinear partial differential equation (PDE) models that are still analytically tractable in most cases but allow explicit consideration of the progression of competitive dynamics as patches age after disturbance. We consider three possible types of density-dependence for their importance in coexistence under disturbance: 1) on reproduction, 2) on recruitment, or 3) on mortality. Although in nature all three types of density-dependence are likely acting in concert, here we consider each in turn on its own, both to help retain analytical tractability, and to identify which type of density-dependence is critical to set up conditions for coexistence.
Under density-dependence on recruitment alone, the model does not permit feasible coexistence. However, under density-dependent reproduction or under density-dependent mortality, variation between two species along a reproduction survival and a sensitivity-to- competition/mortality trade-off both allow for feasible and stable coexistence. One species must have both higher reproduction and higher mortality than the other species in order for coexistence to occur. Inter mediate relative reproduction and death rates, which could arise from intermediate values of disturbance, lead to a wider coexistence region. The results we present contrast importantly with the classical competition-colonization trade-off model by showing that simply a reproduction/survival trade-off can facilitate coexistence under disturbance. This trade-off is arguably more often observed in nature.
The second set of models we consider incorporate size-structure. Models capturing the size-structured nature of competition between trees are complex, and have faced barriers to complete analysis, leaving the specific demographic trade-offs that could allow for coexistence unclear. We study four size-structured nonlinear PDE models which include hierarchical size-structuring while remaining analytically tractable. We again consider three possible types of density-dependence 1) on reproduction, 2) on recruitment or 3) on mortality.
Under only density-dependent mortality we find numerical analysis is needed. While we find that coexistence is not possible under density-dependent recruitment alone, we are able to show that hierarchical size-structuring allows for stable coexistence under density-dependent reproduction when species vary along a reproduction and maximal-height continuum. One species must have a higher reproduction rate while the other is able to grow much taller and escape the harmful effects of competition in order for coexistence to occur. This result provides a more rigorous demonstration of prior results from a more complex model which was analyzed numerically, and also highlights that a trade-off that has been observed empirically among tree species can result in coexistence.
The third set of models we consider employ both size and patch-age structure simultaneously. Size and patch-age structures have been used together through extensive simulation and a variety of trade-offs were found to influence coexistence. We consider a size and patch-age structured model which has both a size hierarchy as well as an explicit patch-age structure. We consider a density-dependence on 1) reproduction or 2) recruitment. Under density-dependent recruitment the model does not permit feasible coexistence. However, under density-dependent reproduction—and when shading effects are felt from all individuals—we show coexistence can occur through a trade-off between growth and survival. (Analysis of a density-dependent reproduction model where shading occurs only from taller individuals is in progress.) The growth/survival trade-off has been widely observed among tree species. The coexistence enabling nature of variation along this trade-off contrasts with forest ecologist’s historical focus on a trade-off between growth in high light and survival in the shade as the key to coexistence.
The results we present in this dissertation are the first (without assuming perfect plasticity or discrete size classes) to show analytical trade-offs resulting in coexistence through models which include hierarchical size-structure, patch-age structure or size and patch-age structure. Overall this dissertation provides a basis for an analytically tractable modeling approach for studying competition between two species (with the potential of extending to multiple species) whose populations are structured by size and/or patch-age and hence provides the core methodology needed for studying coexistence through life-history variation analytically.
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