Relative nonlinearity

Last updated December 09, 2022

Relative nonlinearity is a coexistence mechanism that maintains species diversity via differences in the response to and effect on variation in resource density or some other factor mediating competition. Relative nonlinearity depends on two processes: 1) species have to differ in the curvature of their responses to resource density and 2) the patterns of resource variation generated by each species must favor the relative growth of another species. In its most basic form, one species grows best under equilibrium competitive conditions and another performs better under variable competitive conditions. Like all coexistence mechanisms, relative nonlinearity maintains species diversity by concentrating intraspecific competition relative to interspecific competition. Because resource density can be variable, intraspecific competition is the reduction of per-capita growth rate under variable resources generated by conspecifics (i.e. individuals of the same species). Interspecific competition is the reduction of per-capita growth rate under variable resources generated by heterospecifics (i.e. individuals of a different species). Like some other coexistence mechanisms (see the storage effect), relative nonlinearity can allow coexistence of at least two species on a single resource.

Functional components

Differential nonlinear responses to resources

Relative nonlinearity requires that species differ in the curvature of their fitness response ${\textstyle \phi _{i}}$ to some competitive factor, F, like resource density. The nonlinearity of a response to competition is the second derivative of the per-capita growth rate with respect to the competitive factor ${\textstyle \phi _{i}^{\prime \prime }(F)}$ , which is zero if the growth response is linear, positive if the response is accelerating (convex), and negative if the response is decelerating (concave). For competition between two species, the greater the difference in the curvatures of their response to changes in a competitive factor, the greater the differences in their overall specialization on competitive factor variation. For example, by Jensen's inequality, compared to constant resource density, variation in a competitive factor has no effect on species with zero curvature, positive effects on species with positive curvature, and negative effects on species with negative curvature. Thus, ${\textstyle \phi _{i}^{\prime \prime }(F)}$ indicates a species response to variation in competitive factors, a dimension of competition that can be partitioned.

Competitive factors are best thought of as dimensions of the environment that are jointly used by more than one species and contribute to a reduction in performance of individuals when used. For example, space is a common competitive factor for trees because many species require space for new trees to grow and the reduction in space reduces opportunities for other species to capture that space and grow. Resources and predators have similar properties and count as competitive factors. For competition between two species for a single shared resource, it is easy enough to think of the competitive factor as the reduction in species density due to consumption. In the absence of resource consumption, resources will tend to be at some equilibrium value, K. Thus, the competitive factor for our example is ${\textstyle F=K-R}$ for any value of R.

The original demonstration of relative nonlinearity was in a consumer-resource model with differences in functional responses of the two species. One species has a Type I functional response and has zero curvature. The second species has a Type II functional response - which occurs when individuals must spend time handling resources before moving on to the next resource - and has negative curvature. Because the second species is limited by time when capturing resources, it is unable to exploit resources at high density compared to its competitor. If the Type II functional response species does better under average conditions than the species with a Type I functional response, the species differ in their response to equilibrium and variable resource density.

Differential effect on resource variation

Not only must species respond differently to variation in competition, species must also affect variation in competition differently.

Given these two processes, differential effects on and response to resource variation, species may coexist via relative nonlinearity.

Mathematical derivation

Here, we will show how relative nonlinearity can occur between two species. We will start by deriving the average growth rate of a single species. Let us assume that each species' growth rate depends on some density-dependent factor, F, such that

{\frac {dN_{j}}{dt}}=\phi _{j}(F)N_{j}

,

where N_j is species j's population density, and $\phi _{j}(F)$ is some function of the density-dependent factor F. For example, under a Monod chemostat model, F would be the resource density, and $\phi _{j}(F)$ would be $a_{j}F-d$ , where a_j is the rate that species j can uptake the resource, and d is its death rate. In a classic paper by Armstrong and McGehee ^[1][cite Armstrong], $\phi _{j}(F)$ was the a Type I functional response for one species and a Type II functional response for the other. We can approximate the per-capita growth rate, $r_{j}={\frac {1}{N_{j}}}{\frac {dN_{j}}{dt}}$ , using a Taylor series approximation as

r_{j}\approx \phi _{j}({\overline {F}})+(F-{\overline {F}})\phi _{j}({\overline {F}})'+{\frac {1}{2}}(F-{\overline {F}})^{2}\phi _{j}({\overline {F}})''

,

where ${\overline {F}}$ is the average value of F. If we take the average growth rate over time (either over a limit cycle, or over an infinite amount of time), then it becomes

{\overline {r_{j}}}\approx \phi _{j}({\overline {F}})+{\frac {1}{2}}\sigma _{F}^{2}\phi _{j}({\overline {F}})''

,

where $\sigma _{F}^{2}$ is the variance of F. This occurs because the average of $(F-{\overline {F}})$ is 0, and the average of $(F-{\overline {F}})^{2}$ is the variance of F. Thus, we see that a species' average growth rate is helped by variation if Φ is convex, and it is hurt by variation if Φ is concave.

We can measure the effect that relative nonlinearity has on coexistence using an invasion analysis. To do this, we set one species' density to 0 (we call this the invader, with subscript i), and allow the other species (the resident, with subscript r) is at a long-term steady state (e.g., a limit cycle). If the invader has a positive growth rate, then it cannot be excluded from the system. If both species have a positive growth rate as the invader, then they can coexist.^[2]

Though the resident's density may fluctuate, its average density over the long-term will not change (by assumption). Therefore, ${\overline {r_{r}}}=0$ . Because of this, we can write the invader's density as

{\overline {r_{i}}}={\overline {r_{i}}}-{\overline {r_{r}}}

.^[3] Substituting in our above formula for average growth, we see that

{\overline {r_{i}}}\approx \left(\phi _{i}({\overline {F}})+{\frac {1}{2}}\sigma _{F}^{2}\phi _{i}({\overline {F}})''\right)-\left(\phi _{r}({\overline {F}})+{\frac {1}{2}}\sigma _{F}^{2}\phi _{r}({\overline {F}})''\right)

.

We can rearrange this to

{\overline {r_{i}}}\approx \left(\phi _{i}({\overline {F}})-\phi _{r}({\overline {F}})\right)+\Delta N_{i}

,

where $\Delta N_{i}$ quantifies the effect of relative nonlinearity,

\Delta N_{i}={\frac {1}{2}}\sigma _{F}^{2}\left(\phi _{i}({\overline {F}})''-\phi _{r}({\overline {F}})''\right)

.

Thus, we have partition the invader's growth rate into two components. The left term represents the variation-independent mechanisms, and will be positive if the invader is less hindered by a shortage of resources. Relative nonlinearity, $\Delta N_{i}$ will be positive, and thus help species i to invade, if $\phi _{i}({\overline {F}})''>\phi _{r}({\overline {F}})''$ (i.e., if the invader is less harmed by variation than the resident). However, relative nonlinearity will hinder species i's ability to invade if $\phi _{i}({\overline {F}})''<\phi _{r}({\overline {F}})''$ .

Under most circumstances, relative nonlinearity will help one species to invade, and hurt the other. It will have a net positive impact on coexistence if its sum across all species is positive (i.e., $\Delta N_{j}+\Delta N_{k}>0$ for species j and k).^[4] The $\phi _{j}({\overline {F}})$ terms will generally not change much when the invader changes, but the variation in F will. For the sum of the $\Delta N_{i}$ terms to be positive, the variation in F must be larger when the species with the more positive (or less negative) $\phi _{j}({\overline {F}})''$ is the invader.

Related Research Articles

<span class="mw-page-title-main">Kinetic theory of gases</span> Historical physical model of gases

The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles, all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.

<span class="mw-page-title-main">Unified neutral theory of biodiversity</span> Theory of evolutionary biology

The unified neutral theory of biodiversity and biogeography is a theory and the title of a monograph by ecologist Stephen P. Hubbell. It aims to explain the diversity and relative abundance of species in ecological communities. Like other neutral theories of ecology, Hubbell assumes that the differences between members of an ecological community of trophically similar species are "neutral", or irrelevant to their success. This implies that niche differences do not influence abundance and the abundance of each species follows a random walk. The theory has sparked controversy, and some authors consider it a more complex version of other null models that fit the data better.

In mathematics, a positive-definite function is, depending on the context, either of two types of function.

In mathematics, the total variation identifies several slightly different concepts, related to the structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.

In plasmas and electrolytes, the Debye length $, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/e. A Debye sphere is a volume whose radius is the Debye length. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector for particles of density, charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures are known as the Thomas-Fermi length and the Thomas-Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.$

Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer.

In probability theory and directional statistics, the von Mises distribution is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle $on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation. The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises-Fisher distribution on the N -dimensional sphere.$

The storage effect is a coexistence mechanism proposed in the ecological theory of species coexistence, which tries to explain how such a wide variety of similar species are able to coexist within the same ecological community or guild. The storage effect was originally proposed in the 1980s to explain coexistence in diverse communities of coral reef fish, however it has since been generalized to cover a variety of ecological communities. The theory proposes one way for multiple species to coexist: in a changing environment, no species can be the best under all conditions. Instead, each species must have a unique response to varying environmental conditions, and a way of buffering against the effects of bad years. The storage effect gets its name because each population "stores" the gains in good years or microhabitats (patches) to help it survive population losses in bad years or patches. One strength of this theory is that, unlike most coexistence mechanisms, the storage effect can be measured and quantified, with units of per-capita growth rate.

The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:

Let $(M, g)$ be a closed smooth Riemannian manifold. Then there exists a positive and smooth function $f$ on $M$ such that the Riemannian metric $fg$ has constant scalar curvature.

<span class="mw-page-title-main">Spinodal decomposition</span>

Spinodal decomposition is a mechanism by which a single thermodynamic phase spontaneously separates into two phases. Decomposition occurs when there is no thermodynamic barrier to phase separation. As a result, phase separation via decomposition does not require the nucleation events resulting from thermodynamic fluctuations, which normally trigger phase separation.

<span class="mw-page-title-main">Wrapped normal distribution</span>

In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.

In quantum field theory, a non-topological soliton (NTS) is a soliton field configuration possessing, contrary to a topological one, a conserved Noether charge and stable against transformation into usual particles of this field for the following reason. For fixed charge Q, the mass sum of Q free particles exceeds the energy (mass) of the NTS so that the latter is energetically favorable to exist.

<span class="mw-page-title-main">Wrapped Cauchy distribution</span>

In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.

The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of stochastic processes. Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system.

In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven Ricci scalars which consist of three real scalars $, three complex scalars and the NP curvature scalar . Physically, Ricci-NP scalars are related with the energy-momentum distribution of the spacetime due to Einstein's field equation.$

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

Coexistence theory is a framework to understand how competitor traits can maintain species diversity and stave-off competitive exclusion even among similar species living in ecologically similar environments. Coexistence theory explains the stable coexistence of species as an interaction between two opposing forces: fitness differences between species, which should drive the best-adapted species to exclude others within a particular ecological niche, and stabilizing mechanisms, which maintains diversity via niche differentiation. For many species to be stabilized in a community, population growth must be negative density-dependent, i.e. all participating species have a tendency to increase in density as their populations decline. In such communities, any species that becomes rare will experience positive growth, pushing its population to recover and making local extinction unlikely. As the population of one species declines, individuals of that species tend to compete predominantly with individuals of other species. Thus, the tendency of a population to recover as it declines in density reflects reduced intraspecific competition (within-species) relative to interspecific competition (between-species), the signature of niche differentiation.

The fitness-density covariance is a coexistence mechanism that can allow similar species to coexist because they are in different locations. This effect will be the strongest if species are completely segregated, but can also work if their populations overlap somewhat. If a fitness-density covariance is operating, then when a species becomes very rare, its population will shift to predominantly locations with favorable conditions. Similarly, when a species becomes very common, then conditions will worsen where they are most common, and they will spread into areas where conditions are less favorable. This negative feedback can help species avoid being driven extinct by competition, and it can prevent stronger species from becoming too common and crowding out other species.

In modern valence bond (VB) theory calculations, Chirgwin–Coulson weights are the relative weights of a set of possible VB structures of a molecule. Related methods of finding the relative weights of valence bond structures are the Löwdin and the inverse weights.

The Izbash formula is a formula for the stability calculation of armourstone in running water.

References

↑ Armstrong, Robert A.; McGehee, Richard (February 1980). "Competitive Exclusion". The American Naturalist. 115 (2): 151–170. doi:10.1086/283553. S2CID 222329963.
↑ Schreiber, Sebastian J.; Benaïm, Michel; Atchadé, Kolawolé A. S. (8 June 2010). "Persistence in fluctuating environments". Journal of Mathematical Biology. 62 (5): 655–683. arXiv: 1005.2580 . doi:10.1007/s00285-010-0349-5. PMID 20532555. S2CID 2018289.
↑ Chesson, P. (June 1994). "Multispecies Competition in Variable Environments". Theoretical Population Biology. 45 (3): 227–276. doi:10.1006/tpbi.1994.1013.
↑ Chesson, Peter (November 2003). "Quantifying and testing coexistence mechanisms arising from recruitment fluctuations". Theoretical Population Biology. 64 (3): 345–357. doi:10.1016/s0040-5809(03)00095-9. PMID 14522174.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Armstrong+McGehee-1] Armstrong, Robert A.; McGehee, Richard (February 1980). "Competitive Exclusion". The American Naturalist. 115 (2): 151–170. doi:10.1086/283553. S2CID 222329963.

[schreiber-2] Schreiber, Sebastian J.; Benaïm, Michel; Atchadé, Kolawolé A. S. (8 June 2010). "Persistence in fluctuating environments". Journal of Mathematical Biology. 62 (5): 655–683. arXiv: 1005.2580 . doi:10.1007/s00285-010-0349-5. PMID 20532555. S2CID 2018289.

[Chesson1994-3] Chesson, P. (June 1994). "Multispecies Competition in Variable Environments". Theoretical Population Biology. 45 (3): 227–276. doi:10.1006/tpbi.1994.1013.

[chesson2003-4] Chesson, Peter (November 2003). "Quantifying and testing coexistence mechanisms arising from recruitment fluctuations". Theoretical Population Biology. 64 (3): 345–357. doi:10.1016/s0040-5809(03)00095-9. PMID 14522174.

[1]

[2]

[3]

[4]