Spread of an invasive shrub from Mexico into New Mexico grassland: using reactiondiffusion models

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Spread of an invasive shrub from Mexico into New
Mexico grassland using reaction-diffusion models
Group 3

• Paul Tian, Xiaojie Hou,
• Etsuko Nonaka, Joaquin Rivera, Edgar Diaz, Jaffar
  Ali

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US Southwest

• Desert ecosystems water limited
• Grassland, shrubland, and forests at higher
  elevations
• Important species
• Grassblack grama (Bouteloua eriopoda)
• Shrubcreosotebush (Larrea tridentata)
• mesquite (Prosopis glandulosa)

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Shrub invasion

• Shrubs are expanding their range to northern
  Chihuahuan Desert in New Mexico from Mexico in
  the last 100 to 150 years.
• Historical accounts state that northern
  Chihuahuan desert was good grass 150 years ago
  with some pockets of shrubs in dry areas.
• Shrubs are more common and denser in the Mexican
  part of the Desert.
• The possible reasons 1) drought/climate change,
  2) overgrazing, 3) fire exclusion, etc.
• Drought comes every 50 years (Sheppard et al.
  2002)

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Shrub invasion (contd)

• The problems related to this invasion are
• Loss of grassland loss of forage for cattle
• Sign of desertification
• Reduction of the species diversity
• Changes in water and nutrients resources of the
  soil

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The ecological question

• Will the spread of the shrub species toward north
  stop eventually under current climate conditions?
• Less favorable environment to the shrubs toward
  north
• Grass becomes a better competitor toward north

AZ
NM
?
Mexico
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The ecological question

• Will the spread of the shrub species toward north
  stop eventually under current climate conditions?
• Less favorable environment to the shrubs toward
  north
• Grass becomes a better competitor toward north

AZ
NM
Albuquerque
400 km
Mexico
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Our approaches

• Reaction-diffusion competition model
• Fit data and simulate invasion from 1850 to
  present
• Predict the location of the spreading front 100
  years from now
• Reaction-diffusion pioneer-climax model
• Stability analysis of equilibria

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Some assumptions

• Diffusion of shrubs is only to the north
• Logistic growth
• When shrubs germinate, grass is a competitor
• Natural death rate for adults

• Grass is just invaded (no diffusion)
• Constant growth rate
• Shrubs are stronger competitors (e

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Definition of u, and v

• Percent of area covered by shrubs

Percent of area covered by grass
Here, u 25 v 50
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The model
a diffusion r rate of maximum growth k
carrying capacity µ natural death of adults
(shrubs) e, d interaction of the grass and the
shrubs b constant growth rate (grass)
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The model (contd)
Now, some parameters depend on location!
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The model (contd)
Drought periods matter! Also, time period are
different (5)
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The model (contd)
Grass diffuses? Logistic grow rate in grass?
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Covers of Shrub and Grass
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Shrub
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Grass
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Comparison of Data with Model Prediction
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Pioneer-Climax Model

• Model interactions of species whose reproductive
  capacity is sensitive to population density in
  their shared ecosystem.
• Assumptions of the model
• Pioneer have continuous and monotonically
  decreasing reproductive rates.
• Climax have continuous reproductive rates which
  are continuous, increasing at low average
  population densities and decreasing at high
  densities.
• The species per capita growth rate is a function
  of a weighted total density variable.

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Difference with the Lotka-Volterra Model

• Both fitness functions are pioneer fitnesses,
  purely competition.

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Our Pioneer-Climax Model

• A ecosystem of 2 interacting, continuously
  reproducing populations.
• yi c1i u c2iv (weighted total density) (i
  1, 2)
• cji is called the interaction coefficients.
• yi has the advantage of separating each
  populations response to density, fj , from the
  competitive or cooperative effect of each
  individual interacting population.
• climax species its fitness monotonically
  increases up to a maximum value and then
  monotonically decreases as a function of weighted
  total density.

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Pioneer and climax fitness

• The pioneer fitness function is assume to be
  smooth, monotone decreasing and with a unique
  positive root.
• The climax fitness function is smooth, is one
  humped with two positive roots.

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The equations

• Parameters
• D, e are the diffusion coefficients (for now set
  both equal to 1)
• ai is the growth rates
• Ci interaction coefficient between the two
  species.
• ?i mixed of parameters
• ? death rate for the grass (pioneer species)

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Isoclines for Pioneer-Climax Model
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Equilibrium Points
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Stability Analysis

• Unstable

• Stable

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Numerical Traveling Wavewith D e 1
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Numerical Traveling Wavewith D e
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Numerical Traveling Wavewith D e 1
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Conclusions

• Our numerical model shows that shrubs reach
  Albuquerque (400 km from the border) in 150
  years.
• It predicts that shrubs will invade beyond 500 km
  from the border in another 100 years.
• Some shrubs must have existed in northern
  Chihuahuan desert to achieve the wide spread
  invasion in 150 years
• Areas with the I.C. 0 have to wait until the
  leading front diffuses into the area to build up
  population
• It will talk a long time for shrubs to spread to
  northern NM

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Conclusions

• The mathematical analysis of the pioneer-climax
  model shows that shrubs and grass cannot coexist
• Eventually grass will all die, and shrubs
  dominate as time elapses.
• The density of shrubs needs to be sufficiently
  high for shrubs to persist
• The speed of reproduction needs to be
  sufficiently high

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Challenges

• Parameter estimation of the competition model
• Need reliable data
• Projecting invasion into one dimension
• Heterogeneity of spread and landscape uncertain
• Interdisciplinary cooperation
• Different emphasis

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Future work

• Mathematical analysis of the simple
  reaction-diffusion competition model
• Stage-structured model
• Juveniles and adults have very different
  mortality rates and roles in invasion processes
• Applied questions
• Climate change effects
• Control of shrubs using prescribed fire

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Questions?
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Acknowledgments

• Dr. Yuan Lou
• Dr. Paul (Jianjun) Tian
• All the MBI staffs