Perennial Plant Models to study Species Coexistence in a Variable Environment
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This document presents research on modeling perennial plant species coexistence in a variable environment. It explores how variation in recruitment and individual growth across generations can promote stability of coexistence through different storage effects. It examines how life history tradeoffs between fecundity and growth interact with these storage effects. Key findings include that variation in individual growth can stabilize coexistence, and that asymmetry in how species respond to the environment associated with asymmetry in their average life history traits and population structures strengthens coexistence mechanisms. The models provide a framework for understanding empirical studies on how species' demographic responses to fluctuating conditions allow them to persist together.
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Perennial Plant Models to study Species Coexistence in a Variable Environment
- 1. Perennial Plant Models to study Species Coexistence in a Variable Environment Chi Yuan
- 4. Coexistence in variable environment • Species coexistence – Using environment differently – Puzzling, plants share similar resources • Variable environment – Difference in efficiency in using the resource • Which life history processes vary with environment? Year 2Year 1
- 5. • Which life history processes vary with environment? – Species-specific responses in recruitment Chesson et al. 2013
- 6. • Which life history processes vary with environment? – Species-specific responses in individual growth Enquist and Leffler, 2001, Long-term tree ring chronologies from sympatric tropical dry-forest trees: individualistic responses to climatic variation
- 7. Outline of the research
- 8. Non-structure lottery model: recruitment variation only Reproduction as Environmental Response Establishment as Competitive Response Survival A definition of Lottery competition Picture credit: http://de.sap.info/wp-content/uploads/2013/02/SME_Growing_Plant_iStock.jpg http://bestclipartblog.com/25-tree-clip-art.html/tree-clip-art-1
- 9. • Stability of coexistence – Invasibility analysis • Resident • Invader – Stabilizing effect – Equalizing effect ri = xi + A A I N Residents invader
- 10. • Storage effect – Covariance – Buffer • Survival Better environment Strongercompetition
- 11. Relative nonlinearity • Relative nonlinear growth rates in responses to competition Longer-lived species is favored by larger fluctuation in competition Shorter-lived species is favored by larger fluctuation in competition
- 12. Life history characters affecting both mechanisms • Difference in death rate – Fecundity-mortality tradeoff • Difference in sensitivity to environment • Correlation in environmental responses 0 10 20 30 40 50 60 70 80 90 100 -8 -6 -4 -2 0 2 time environmentalresponses species 1 species 2 0.55 0.6 0.65 tionsize species 1 species 2 0 10 20 30 40 50 60 70 80 90 100 -8 -6 -4 -2 0 time environmentalresponses species 1 species 2 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.35 0.4 0.45 0.5 0.55 0.6 0.65 time populationsize species 1 species 2 Equal sensitivity, zero correlation Different sensitivity, full correlation
- 13. Sensitivity differences between species Wright 2005
- 14. Storage effect and relative nonlinearity No correlation between speciesHigh correlation between species
- 15. Part 1, recruitment variation: summary • Relative nonlinearity is more restrictive compared with storage effect • Relative nonlinearity compensates the weakening storage effect • Differences in death rate have a big affect on coexistence mechanisms only when aligned with sensitivity differences
- 16. Outline of the research
- 17. size-structured lottery model • Introducing the continuous size structure – Explicit post- recruitment dynamics – Size dependency in demographic rates • Difference from other forest models Age or size Fecundity Age or size Mortality
- 18. Lottery competition in structured model ( ) ( ) ln ( ) R t C t A t ( ) ( ) , ( ) { ( ) ( ) [ ( ) ] ( ) }gi gi bjE E t E t j jc j jc js j js js j jc jc c j R t e c a s a a c a e c k a e N ( ) , ( ) (1 ( ))jc j jc t c j A t N s a Resource needed Resource supply
- 19. Cohort Based models: How environment and competition affect the critical life history process • Eb environmental response in recruitment • Eg environmental response in growth. ( ) ( ) ( ) seedling growth seedling establishment ( ) ( ) bj gi E t js j jc c E t js j js C t e e c k a a c a e Year 1 Year 2
- 20. Cohort based model: How environment and competition affect the critical life history process • Tree growth for cohort c ( ) ( ) ( ) ( ( )) gjE t jc j jc C t e a t c a t e ( )1 ( )j jc ts a ( )( )j jc ts a ( )jca t Size at time t Size at time t+1
- 21. Model overview Seedling establishment Tree growth )( () ( 1) bj jc E C t jn js a jc t c k eN t c N ( ) ( 1) gj js E jn js a t ca et a Seedling Growth ( ) ( ) ( )( 1) ( ) gj jc E jc jc a t C t tca a tt e
- 22. Two species • Species are identical on averages, except their responses to the environment
- 23. Variation in individual growth can also promote species coexistence thestabilizingeffect 0.00.20.40.60.81.01.21.4 Eb Eg Eb+Eg Eb Eg Eb+EgEb Eg Eb+Eg Eb Eg Eb+Eg Fecundity increases slower with size Fecundity increases faster with size • Recruitment variation Vs. variation in individual growth • Storage effect as the major contributor • Two form of storage effect
- 24. Interaction between reproduction and individual growth – Variation in reproduction and growth is not additive • Synergism when growth and reproduction is positively related • Antagonisms with negative correlation A ∆𝐼
- 25. Interaction between reproduction and individual growth A ∆𝐼
- 26. Two form of storage effect • Mean fractional contribution to population growth 𝑝 𝑏 + 𝑝 𝑏𝑔 + 𝑝 𝑔 + 𝑝𝑠 = 1
- 27. pb pb
- 29. How life history affects the storage effects • An example increasing pb and Pbg by increasing seedling size A ∆𝐼 A ∆𝐼 Variation in reproduction only Variation in growth only
- 30. The effect of shapes in demographic schedules • Changes in overall stabilizing effect • Storage effect is not sensitive to shapes of the schedules, given p’s are fixed a. b. A ∆𝐼 A ∆𝐼
- 31. Shift in size structure • Variation in growth: more larger individuals in invader state than resident state • Variation in reproduction: more smaller individuals in invader state than resident state (b) Eb only (a) Eg only Invader state Resident state Resident state Invader state
- 32. Effect of shift in size structure • ΔS Mean structure effect under equilibrium environment • ΔE Changes in mean environment effect due to shift in structure (a) Eg only Invader state Resident state Being smaller more advantageous Being larger more advantageous
- 33. Effect of shift in size structure • ΔS Mean structure effect under equilibrium environment • ΔE Changes in mean environment effect due to shift in structure (b) Eb only Resident state Invader state Being smaller more advantageous Being larger more advantegous
- 34. Effects of shapes in demographic schedule through shifts in structure ref flat f flatm flatc mechanismpartition 0.0 0.2 0.4 0.6 E Cs N ref flat f flatm flatc mechanismpartition -0.5 0.0 0.5 E Cs N Eb only Eg only ref ref ref Flat f Flat m Flat c
- 35. Part two: summary • General theory is compatible with studies of interesting biological details • Variation in individual growth promote coexistence • Storage effect as the main stabilizing mechanisms – Only the relative contributions of key processes to population growth in a population matters for storage effect • Storage effect is strong when processes most sensitive to environment also contribute most strongly on average to population growth • The effect of size-dependency in life history is determines by shift in structure
- 36. Outline of the research
- 37. Part 3 Life history tradeoff • Difference in life history strategy between species – Formulated as tradeoff • Tradeoff and species coexistence – Equalizing effect – Stabilizing? Wright et al. 2010 Jakobsson and Eriksson, 2000 ri = xi + A
- 38. Case 1 • Tradeoff between fecundity and growth – Species 1 with mean advantage in reproduction (solid) – Species 2 with mean advantage in growth (dash) sp1 sp1sp2 sp2 Identicalaverage average
- 39. In constant environment • Equalizing effect of tradeoff in mean demographic properties – No stabilizing effect alone Difference in mean environmental responses sp1 sp2 Sp1 winsSp2 wins
- 40. Fluctuation dependent mechanisms • Stable coexistence – ΔS • Mean structure effect – ΔE • Mean environment effect – ΔI • Covariance between environment and competition • Buffer • Stabilizing effect • Fitness inequality ri = xi + A i i i iS E S I A E I i i i i i i S S S E E E I I I
- 41. In variable environment • Variation in reproduction – Species 1 with mean advantage in reproduction – Species 2 with mean advantage in growth • Equalizing effect of the tradeoffs – Compensating between dE and dI • Small effect of shift in structure Strongly asym Sym δI1b δI2b δE1b δE2b δS1b δS2b
- 42. In variable environment • Variation in reproduction • Stabilizing effect – Storage effect as the main stabilizing mechanism Strongly asym Sym ∆𝐼 ∆E ∆S
- 43. Alignment between sensitivity and tradeoff • Species with mean advantage in fecundity (sp1) has fecundity more sensitive to environment, species with mean advantage in individual growth (sp2) has growth more sensitive to environment
- 44. Case 1 • Tradeoff in population average properties • No significant effect of shift in structure sp1 sp1sp2 sp2 Identicalaverage average
- 45. Case 2 • Ontogenetic tradeoff – An extreme case where shift in structure have bigger effect – two species have contrasting shapes of demographic schedules sp1 sp1 sp1 sp2 sp2 sp2
- 46. Asymmetry in sensitivity and shapes • Sp1: being small has more demographic advantage • Sp2: being large has more demographic advantage • Sp1 has only reproduction varies, sp2 has only growth varies sp1 sp2 community average mechanismpartition 0.0 0.2 0.4 0.6 S E Cs N I
- 47. Part 3 Summary • Tradeoff in demographic traits alone only have equalizing effect • Tradeoff interact with equalizing effect of the fluctuating dependent mechanisms • Importance in asymmetry in sensitivity associated with asymmetry in mean life history traits. • Contrast in population average properties, and shapes of demographic schedules
- 48. Implication • Quantification methods apply in general • Multiple coexistence mechanisms interacting – Some assumptions holds more easily, others more restrictive • Life history traits are good predictors of the strength of the mechanisms • Variation in recruitment and variation in growth • Tradeoffs and sensitivity difference in environmental responses
- 49. Acknowledgment Advisor: Peter Chesson Committee: Judie Bronstein Mike Rosenzweig Larry Venable Jim Cushing Brian McGill The Lab: Galen Holt, Yue (Max) Li, Pacifica Sommers, Simon Stump, Nick Kortessis, Jessica Kuang, Danielle Ignace, Lina Li, Andrea Mathias, Stephanie Hart, Krista Robinson, Elieza Tang EEBer: Guan-Zhu Han, Jin Wu, Ginny Fizpatrick, Sara Felker, Jonathan Horst, Lindsey Sloat, Will Driscoll, Xingyue Ge Liz Oxford, Lili Schwartz, Carole Rosenzweig Barry McCabe, Sky Dominguez, Lauren Harrison, Pennie Liebig Friends Ding Ding, Muhua Wang, Muhan Zhou, Rick and Linda Hanson Family Ying Yu and Jianzhong Yuan Li Fan Funding source Science Foundation Arizona NSF research assistantship EEB department GPSC Institute of Environment HE Carter travel award
Editor's Notes
- http://andrebaertschi.photoshelter.com/image/I0000uBMEeCkN0Jw Rio Tuichi, Madidi National Park, La Paz, Bolivia. A good picture to raise big question Love nature for aesthetic reason, but then I learn there is more than that Study ecology helps me to increase the awareness to recognizing the diversity of species in nature
- In any system, either in forest or in desert, there are a high diversity of species. Keeps us wondering how is the diversity maintained,
- Species coexistence is a central topic in ecology There are many potential hypothesis, but many are conflicting, very few testable, or quantifiable. Stable coexistence arises when species use environment different. Here is a three level trophic chain. If we study the coexistence of two focal species, we want to know they use their resources differently. There is a symmetric effect from predators. But we will focus on competition. Species coexist when they partition their resource use. There is a density feedback loop that if a species draw down the resource they use most efficiently, they post strong self limitation. Focus on plants: For plants it is a bit more puzzling, they share similar resource requirement, water, nutrient, co2. If we look at this simplified diagram for plants, where they really have similar requirement for resources. We may ask how they partition in resource. People usually seek the answer from fluctuating physical environment. Under different environment, species are active in different time in resources use. Species 1 have an advantages in using the resources in one year, sp2 have the advantage in another year, no species is going to perform uniformly better than the other, species coexistence.
- This part betters goes to the other part to explain the sensitivity differences Indeed species are respond to the variation in different ways. critical life history process of plants varies, Here is a example from winter annual community in Chihuahua desert, fluctuation in abundances of different species are plotted with time. we see a different species are favored in different years. Lab experiment further show that these annual plants shows these species show distinct responses in temperature. Some germinated best in low temperature, some germinated best in high temperature. Another example of high recruitment variation is from three different forests, here the distribution of the magnitude of variation is plotted. We see the variation, measured by the coefficient of variation, are quite large. some species are far more sensitive to environment than other species.
- Remind people about variation in individual growth in part 2, one graph in growth variation While recruitment can be most variable, other process, such as individual growth of tree, also varies. Here is an example for variation in growth of 6 tree species in 8 years, measured from deviation from the mean. tree ring holds longer records for tree growth. There is a more obvious contrast in pattern of individual growth between two species in the tropical dry-forests. Questions arises how important are these variations in different life history process for species coexistence. have contrasting Possible [Putative] adaptation to the changing environment?
- Explain how these three are linked Emphasizing the last part is about different species has different importance of recruitment and growth Put research questions heres. Rather than the component of models
- Remind the research questions, the role of recruitment variation The theoretical understanding of how variable environment shape community dynamics is build on a simple, non structure lottery model. In this model, adult produce seeds, this process is sensitive to environmental variation. seedling compete to establish as adult. The model is named lottery model because the competition. If one species is taking advantage of the environment and have a better environmental responses than the other species, which means it produced more seeds, it will also have larger number of seedling win the competition and get established. However, these simple model is only able to capture the reproduction process, it is not covering the process of growth. Unlike seedling recruitment that just happen in the very beginning of life history, growth take up of the majority part of the life span. It is interesting to ask whether partitioning environment during growth is also helpful
- Recover of invader species in the presence of residents, multi species Highlight the mechanisms when explaining Add the slides show how relative nonlinearity works. Explain buffer Tell people i did not include the details for approximation for mechanisms, ask me later if you are interested
- Arrow, better environment Higher competition
- Subtitle showing the differences between two figures Extreme comparisons
- Titles saying these are the critical life history characters that affect both mechanisms. Explain why changing these variables. Add the slides show how relative nonlinearity works. Put the empirical evidence for sensitivity differences before this slides Subtitles for the graphs Explain buffer Tell people i did not include the details for approximation for mechanisms, ask me later if you are interested
- Mean death rate and mean sensitivity is fixed constant Better link between 1part and 2part Two bullets, relative nonlinearity more restricted, Emphasize the effect of correlation
- Summary can be used as transition to the second part More informative summary, key results Relative nonlinearity is more restricted therefore ignored in later chapters Though relative nonlinearity is limited, it can be compensating for cases when storage effect is weak As a bridge to the next session Asymmetry in mechanism?
- Back to the flow chart to remind big structure and introduce the next questions
- Dual meaning parts does not make sense, make it more explicit Put growth schedule here Size is an important life history traits, also important for formulation of other life history strategies Its role in life history evolution and demography, allow formulation of plant strategies Reproduction, germination and individual growth may pick up different environment cues
- Some individual die, not some species die Here is a graphical representation of how the model works. The equations for the cohort based model is used here for example, but you don’t need to read the equations to understand the talk. so I will walk through the equations briefly. In a community there are trees of different species represented by different colors, and different sizes. Each year, some tree are dead and released the resources, represent by A(t) which is sum over source released by different cohort of different species. This is the resource supply. The potential requirement for space is Rt, which has two part, the established individual is trying to get larger, and also there are new seedling that has not shown here on the graph yet, trying to establish. Resource need is usually much larger than resource supply. Competition responses is defined as the ratio of the need to the supply. We take the log of this ratio.
- Picture credit: http://de.sap.info/wp-content/uploads/2013/02/SME_Growing_Plant_iStock.jpg
- Growth is the other process in the model. For A given tree in a given cohort with size ajc(t) at time t, it will have a certain probability to die. The mortality is size dependent. If it survive, it will grow to a larger size. The growth is proportion to environmental response in growth and inversely proportion to the competitive response. Of course, it is also proportion to the competitivness of the individual at that size. To added in complexity, you can have different environmental responses in growth between seedling and adult, or between different sizes of adult, but for now, I kept them the same.
- Don’t show the dashed line Environmental responses are different between species Shows the fluctuations of environmental responses Flat lines are averages of the size-dependent ones, but it is weighted more heavily with range of the smaller individuals Solid lines are the references
- One curves for setting one, one curves for setting two
- Covariances? Highlight that storage effect is the main coexistence mechanism Walk through the tables Tell you the what results in the difference between storage effect and A
- Covariances? Point out a region on the graphs that highlights the regions on the left graph Remind people A is mainly contributed by the storage effect Walk through the table about the main components the storage effect On top of the graph, point out positive correlation, or negative ones
- Point out theta b and theta g are in both covEC terms , all the structure reduces to fractions of different classes Have an equations shows that Fractions sums to one
- Storage effect is strongest when the process most sensitive to environment is also on average contribute a lot to population growth
- The effect of life history details on the storage effect can be summarize by four critical parameters, fractional contribution of survival, seedling establishment, growth during recruitment, growth after recruitment. As these fractions sums up to one, really three of them matters. Fraction in survival affect both storage effect in the same way, what determines relative importance of the two storage effect are the relative importance of pb Pbg and Pg Whether processes most sensitive to environment also contribute most strongly to competition on average, and thus contribute more to population growth
- Put labels on variation in reproduction and variation in growth Don’t need panel c and d, or animate them Natural follow of the previous graphs, increasing p_b relative to p_g
- Don’t show the bottom figures
- Shift in size structure as a consequence from storage effect rather than an input into the model Put delta S and Delta E on a separate graphs Shape of demographic schedules affect species coexistence because the shift in mean structure when a species drops to low density. Though the effect is usually much weaker than storage effect. In this cases, shift in structure is a consequence of the low density advantages of the storage effect. variation in reproduction lead to more smaller sized individuals in invader states compared with the resident states because invader has the low density advantage in recruitment, simply produce more smaller individuals. Variation in growth works similarly. How does the shift promote species coexistence? Shift in size structure only promote coexistence if shift in direction with more demographic advantages. If being small has more demographic advantage, shift towards smaller size under variation in reproduction is beneficial. If being large has more demographic advantage, it is the other way around. For ΔE, the issue is whether the structure difference between invader and resident gives greater benefits of the variable environment to the invader Low density fitness advantage if being small are equal or more advantages in fecundity or mortality Low density fitness advantages if being large are more advantages in fecundity and mortality
- Shift in size structure as a consequence from storage effect rather than an input into the model ΔE Mean effect of variable environment due to shift in structure [shift in structure changes the overall population level responses to environment] Put delta S and Delta E on a separate graphs Shape of demographic schedules affect species coexistence because the shift in mean structure when a species drops to low density. Though the effect is usually much weaker than storage effect. In this cases, shift in structure is a consequence of the low density advantages of the storage effect. variation in reproduction lead to more smaller sized individuals in invader states compared with the resident states because invader has the low density advantage in recruitment, simply produce more smaller individuals. Variation in growth works similarly. How does the shift promote species coexistence? Shift in size structure only promote coexistence if shift in direction with more demographic advantages. If being small has more demographic advantage, shift towards smaller size under variation in reproduction is beneficial. If being large has more demographic advantage, it is the other way around. For ΔE, the issue is whether the structure difference between invader and resident gives greater benefits of the variable environment to the invader Low density fitness advantage if being small are equal or more advantages in fecundity or mortality Low density fitness advantages if being large are more advantages in fecundity and mortality
- Shift in size structure as a consequence from storage effect rather than an input into the model ΔE Mean effect of variable environment due to shift in structure [shift in structure changes the overall population level responses to environment] Put delta S and Delta E on a separate graphs Shape of demographic schedules affect species coexistence because the shift in mean structure when a species drops to low density. Though the effect is usually much weaker than storage effect. In this cases, shift in structure is a consequence of the low density advantages of the storage effect. variation in reproduction lead to more smaller sized individuals in invader states compared with the resident states because invader has the low density advantage in recruitment, simply produce more smaller individuals. Variation in growth works similarly. How does the shift promote species coexistence? Shift in size structure only promote coexistence if shift in direction with more demographic advantages. If being small has more demographic advantage, shift towards smaller size under variation in reproduction is beneficial. If being large has more demographic advantage, it is the other way around. For ΔE, the issue is whether the structure difference between invader and resident gives greater benefits of the variable environment to the invader Low density fitness advantage if being small are equal or more advantages in fecundity or mortality Low density fitness advantages if being large are more advantages in fecundity and mortality
- To give an example of how shape of demographic schedules affect species coexistence, I compared the reference demographic schedules with the flat ones, while fixing their life-time average property the same. Storage effect->shift in structure Where the shape has an effect, but can be limited Promote or undermine coexistence, shape comparison
- Change it to the main findings
- Back to the flow chart to remind big structure and introduce the next questions
- Confusions and mixed-up: Demographic niches, Evidence for and against FIG. 1. The growth–mortality trade-off for saplings expressed as the 95th percentile relative growth rate (RGR95) vs. the mortality rate of the slowest growing 25% of individuals (M25) for 103 tree species with mean maximum height . 5 m on Barro Colorado Island, Panama. Open, shaded, and solid symbols represent species with factor scores in the top, middle, and bottom thirds, respectively, for the first principal component of the RGR95–M25 trade-off. Minimum sample sizes are 100 trees/species for M25 and 333 trees/species for RGR95. 2. (a) The relationship between log seed number per individual and log seed weight in 72 plant species inhabiting semi-natural grasslands (r0.55, pB0.05).
- Emphasize the difference between species Remind about the equalizing and stabilizing effect somewhere
- Put a vertical line, where at each side, one species get excluded
- Equalizing diminish xi, stabilizing effect increase A
- Put these into supplementary The net effect is equalizing, Interact with variation in demographic properties , plot fitness inequality and the community average effect separately? In Two species cases, symmetric cases seems to be predictive of the asymmetric cases Change the line color of the plots
- Interact with variation in demographic properties , plot fitness inequality and the community average effect separately? In Two species cases, symmetric cases seems to be predictive of the asymmetric cases Change the line color of the plots
- Interact with variation in demographic properties , plot fitness inequality and the community average effect separately? In Two species cases, symmetric cases seems to be predictive of the asymmetric cases
- Interact with variation in demographic properties , plot fitness inequality and the community average effect separately? In Two species cases, symmetric cases seems to be predictive of the asymmetric cases
- Not the tradeoff contributing to stable coexistence, but whether sensitivity is allign Sensitivity means variance in environmental responses
- Emphasize the difference between species Remind about the equalizing and stabilizing effect somewhere
- Fixed the labels of the graphs
- What the finding means for field studies Looking for tradeoff alone doesn’t help, need to know how sensitivity are associated with the tradeoff Identify important lifehistory as empirical implication