Dynamical System of a Two Dimensional
Stoichiometric Discrete Producer-Grazer
Model : Chaotic, Extinction and Noise
Effects
Yun Kang
Work with Professor Yang Kuang and
Professor Ying-chen Lai ,
Supported by Professor Carlos Castillo-Chavez
(MTBI) and Professor Tom Banks (SAMSI)
Outline of Today’s Talk
 Introduce LKE – model, and its corresponding discrete
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case;
Mathematical Analysis: bifurcation study
Biological Meaning of Bifurcation Diagram;
Chaotic behavior and Extinction of grazer;
Nature of Carry Capacity K and Growth Rate b, and
their fluctuation by environments: adding noise
Interesting Phenomenal by adding noise: promote
diversity of nature
Conclusion and Future Work
Stoichiometry
 It refers to patterns of mass balance in chemical
conversions of different types of matter, which often
have definite compositions
most important thing about stoichiometry
 we can not combine things in arbitary proportions; e.g.,
we can’t change the proportion of water and dioxygen
produced as a result of making glucose.
 Energy flow and Element cycling are two
fundamental and unifying principles in ecosystem
theory
 Using stoichiometric principles, Kuang’s
research group construct a two-dimensional
Lotka–Volterra type model, we call it LKEmodel for short
Assumptions of LKE Model
 Assumption One: Total mass of phosphorus in the
entire system is closed, P (mg P /l)
 Assumption Two:
Phosphorus to carbon ratio (P:C) in the
plant varies, but it never falls below a minimum q (mg P/mg C); the
grazer maintains a constant P:C ratio, denoted by
(mg P/mg C)
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 Assumption Three:
All phosphorus in the system is divided
into two pools: phosphorus in the plant and phosphorus in the
grazer.
Continuous Model
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p is the density of plant (in milligrams of carbon per liter, mg C/l);
g is the density of grazer (mg C/l);
b is the intrinsic growth rate of plant (day−1);
d is the specific loss rate of herbivore that includes metabolic losses (respiration)
and death (day−1);
 e is a constant production efficiency (yield constant);
 K is the plant’s constant carrying capacity that depends on some external factors
such as light intensity;
 f(p) is the herbivore’s ingestion rate, which may be a Holling type II functional
response.
Biological Meaning of Minimum Functions
P  g
min( K ,
)
q
 K controls energy flow and (P −
y)/q is the carrying capacity of
 determined by
the plant
phosphorus availability;
 e is the grazer’s yield constant,
e min( 1,
( P  g ) / p

)
which measures the conversion
rate of ingested plant into its
own biomass when the plants
are P rich ( 1  ( P  g ) / p ); If the
plants are P poor ( 1  ( P  g ) / p ),
then the conversion rate suffers
a reduction.
Continuous Case:
b=1.2 and b=2.9
Discrete Model From Continuous One
 Motivation: Data collect from discrete time, e.g., interval
for collecting data is a year.
 Biological Meaning of Parameters : Modeling the
dynamics of populations with non-overlapping generations
is based on appropriate modifications of models with
overlapping generations.
 Choose
cp
f ( p) 
a p
Mathematical Analysis
 We study the local stability of interior
equilibrium E*=(x*,y*)
Bifurcation Diagram and Its Biological
Meaning
 For continuous case: K=1.5
Bifurcation Diagrams on Parameter b
Bifurcation Diagrams on Parameter b
Bifurcation Diagrams on Parameter K
Relationship Between K and b:
 From these figures, we can see that there is
nonlinear relationship between K and b
which effect the population of plant and
grazer:
For bifurcation of K, increasing the value of b,
the diagram of b seems shrink.
For bifurcation of b, increasing the value of K,
bifurcation diagram seems move to the left
Extinction of Grazer
 From bifurcation diagram, we can see that
for some range of K and b, grazer goes to
extinct. What are the reasons?
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Basin Boundary For Extinction
Global Stability Conjecture
 We know that Discrete Rick Model :
x(n+1)=x(n)exp{b(1-x(n)/K)} has global stability
for b<2, does our system also has this properties
More general, if we have u(n+1)=u(n)exp{f(u(n),0}
with global stability, then the following discrete
system:
x(n+1)=x(n)exp{f(x(n),y(n))+g(x(n),y(n))}, of g(x,y)
goes to zero as y tending to zero, in which
condition has global stability
Nature of K and b
 K is carrying capacity of plant, and it is
usually limited by the intensity of light and
space. Since K is easily affected by the
environment, it will not be always a
constant ;
 b is maximum growth rate of plant, it will
fluctuates because of environment
changing.
Adding Noise
 Because of the nature of biological meaning
of K and b, it makes perfect sense to think
these parameters as a random number.
 We let K=K0+ w*N(0,1)
b=b0+w*N(0,1)
Then Most Interesting thing on
parameter K :
 Prevent extinction of grazer :
Time Windows
Scaling
 Define the degree of
existence :
R=average population of
graze/ average
population of plant
Then try different
amplititute of noise w,
then do the log-log
scaling, it follows the
scaling law.
Future Work
 We would like to use “snapshot” method to see
how noise effects the population of grazer and
producer;
 Try to different noise, e.g. color noise, to see how
the ‘color’ effect the extinction of the grazer;