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November 24, 2003 (^) Start Presentation
Population Dynamics II
• In this class, we shall analyze behavioral
patterns of ecosystems, in which more than
two species interact with each other.
• Such systems frequently exhibit chaotic
behavior.
• Chaotic models shall be analyzed and
discussed.
November 24, 2003 (^) Start Presentation
Table of Contents
- Generalization of ecological models
- Gilpin model
- Chaotic motion
- Bifurcations
- Structural vs. behavioral complexity
- Limits to complexity
- Forces of creation
November 24, 2003 (^) Start Presentation
Generalization of Ecological Models
- What happens when e.g. three species compete for the same food source? The previously used model then needs to be extended as follows:
- There never show up terms such as:
- as such a term would indicate that e.g. the competition between x 2 and x3 disappears, when x 1 dies out.
x· 1 = a · x 1 − b 12 · x 1 · x 2 − b 13 · x 1 · x 3 − b 23 · x 2 · x (^3) x· 2 = c · x 2 − d 12 · x 1 · x 2 − d 13 · x 1 · x 3 − d 23 · x 2 · x (^3) x· 3 = e · x 3 − f 12 · x 1 · x 2 − f 13 · x 1 · x 3 − f 23 · x 2 · x (^3)
− k · x 1 · x 2 · x (^3)
November 24, 2003 (^) Start Presentation
The Gilpin Model I
- Michael Gilpin analyzed the following three-species ecosystem model:
- A single predator, x 3 , feeds on two different species of prey, x 1 and x2 , both of which furthermore compete for the same food source, and suffer from crowding effects.
- The initial populations for all three species were arbitrarily set to 100 animals each. We simulate over 5000 time units.
x· 1 = x 1 − 0.001 · x 12 − 0.001 · x 1 · x 2 − 0.01 · x 1 · x (^3) x· 2 = x 2 − 0.0015 · x 1 · x 2 − 0.001 · x 22 − 0.001 · x 2 · x (^3) x· 3 = - x 3 + 0.005 · x 1 · x 3 + 0.0005 · x 2 · x (^3)
November 24, 2003 (^) Start Presentation
The Gilpin Model IV
November 24, 2003 (^) Start Presentation
The Gilpin Model V
- Most of the time, there are plenty of x 2 animals around.
- Once in a while, the predator ( x3 ) population explodes in a pattern similar to that of the Lotka-Volterra model.
- The predator then heavily decimates the x 2 population.
- The x1 population is usually hampered by strong competition from the x 2 population for the common food source.
- Thus, when the x 2 population is decimated, the x 1 population can thrive for a short while.
- However, the x 2 population recovers quickly, depriving again the x 1 population of their food.
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The Gilpin Model VI
- Yet, the behavioral pattern of each cycle is slightly different from that of the previous one. This can be better seen in phase portraits.
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The Gilpin Model VII
- In a limit cycle , the phase portrait would show a single orbit.
- The observed behavior is called chaotic. Each orbit is slightly different from the last. If the simulation were to proceed over an infinite time period, the orbits would cover an entire region of the phase plane.
- Chaotic behavior is caused here, because the two preys can coexist at different equilibrium levels, i.e., the predator can be fed equally well by eating animals of the x 1 kind as of the x 2 kind. One prey can substitute the other.
- In continuous-time systems , chaos can only exist in 3 rd and higher order systems.
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The Discrete-Time Logistic Equation III
In the range a ∈ [1.0, 3.0] , there are two intersections between the two functions.
The iteration converges rapidly for intermediate values, but as a approaches either a value of a = 1.0 or alternatively a value of a = 3.0 , the iteration converges more and more slowly.
However, only one of the two solutions is stable. There is still only one steady-state solution.
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The Discrete-Time Logistic Equation IV
In the range a ∈ [3.0, 3.5] , a limit cycle is observed.
For a = 3.05 and a = 3.3 , the discrete limit cycle has a period of 2.
For a = 3.45 and a = 3.5 , the discrete limit cycle has a period of 4.
November 24, 2003 (^) Start Presentation
The Discrete-Time Logistic Equation V
In the range a ∈ [3.5, 4.0] , the observed behavioral patterns become increasingly bizarre.
For a = 3.56 , a discrete limit cycle with a period of 8 is being observed.
For a = 3.6 , the behavior is chaotic.
For a = 3.99 , the behavior is again chaotic.
For a = 3.84 , a discrete limit cycle with a period of 3 is being observed.
For a > 4.0 , the system is unstable.
November 24, 2003 (^) Start Presentation
The Discrete-Time Logistic Equation VI
- We can plot the stable steady-state solutions as a function of the parameter a.
The dark region in the plot to the left is the chaotic region , yet, even within the chaotic region, there are a few non- chaotic islands , such as in the vicinity of a = 3..
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Bifurcations III
- We linearize this difference equation around the origin, and find:
- This difference equation is marginally stable for a = 1. and a = 3..
- We now proceed assuming a stable limit cycle with a discrete period of 2, thus:
ξ k+1 = (2.0 – a ) · ξ k
x (^) k+2 = a · x (^) k+1 · (1.0 – x (^) k+1 ) ≡ x (^) k ; k → ∞
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Bifurcations IV
- We evaluate this equation recursively, until x (^) k+2 has become a function of xk only.
- This leaves us with a 4 th^ order polynomial in xk.
- The previously found two solutions must also satisfy this new polynomial, i.e., we can divide by these two solutions, and again obtain a 2 nd^ order polynomial in xk.
- This new polynomial has again two solutions. One of them is a = 3.0 , the other provides us with the next bifurcation point.
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The Gilpin Model VIII
- Let us now look once more at the Gilpin model. We shall treat the competition factor k as the parameter to be varied in the experiment:
- The nominal value of k is k = 1..
- We shall vary k around its nominal value.
- We shall display only the x1 population.
x· 1 = x 1 − 0.001 · x 12 − k · 0.001 · x 1 · x 2 − 0.01 · x 1 · x (^3) x· 2 = x 2 − k · 0.0015 · x 1 · x 2 − 0.001 · x 22 − 0.001 · x 2 · x (^3) x· 3 = - x 3 + 0.005 · x 1 · x 3 + 0.0005 · x 2 · x (^3)
November 24, 2003 (^) Start Presentation
The Gilpin Model IX
For k = 0.98 , we observe a limit cycle with peaks reaching each time the same level. For k = 0.99 , we observe a limit cycle where the peaks toggle between two discrete levels. Only looking at the peaks, we could say that we have a limit cycle with a discrete period of
For k = 0.995 , we have a limit cycle with a discrete period of 3. For k = 1.0 , the behavior is chaotic.
November 24, 2003 (^) Start Presentation
True Behavior or Numerical Artifact II?
- To this end, I propose to apply a logarithmic transformation on the Gilpin model:
- The modified Gilpin model presents itself as follows:
- The analytical results of the two models must be identical, yet their numerical properties are very different.
yi = log (x (^) i )
y· 1 = 1.0 − 0.001 · exp (y 1 ) − 0.001 · exp (y 2 ) − 0.01 · exp (y 3 ) y· 2 = 1.0 − 0.0015 · exp (y 1 ) − 0.001 · exp (y 2 ) − 0.001 · exp (y 3 ) y· 3 = -1.0 + 0.005 · exp (y 1 ) + 0.0005 · exp (y 2 )
November 24, 2003 (^) Start Presentation
True Behavior or Numerical Artifact III?
- We can now plot the discrete bifurcation maps of the two models. If they are the same, then chaos is indeed for real also in this model.
Original Gilpin model
Modified Gilpin model
November 24, 2003 (^) Start Presentation
Structural vs. Behavioral Complexity I
- We have seen that simple deterministic differential equations can lead to incredibly complex behavioral patterns in the solution space.
⇒ The behavioral complexity of a system is generally
much greater than its structural complexity.
Structure Behavior x = -a · x· x(t = 0.0) = x 0 x(t) =^ exp (-a^ ·t)^ ·^ x^0 linear exponential
Structure Behavior
Gilpin model Population trajectories deterministic chaotic
November 24, 2003 (^) Start Presentation
Structural vs. Behavioral Complexity II
- Looking at the Gilpin model , we may reach the conclusion that chaotic behavior is the exception to the rule, that it occurs rarely, and is rather fragile.
- Nothing could be farther from the truth.
- As the structural complexity (the order of a differential equation model) increases, the chaotic regions grow larger and larger. In fact, they quickly dominate the overall system behavior.
- It is thus utterly surprising that no-one recognized chaos for what it is until the 1960s. Before then, chaotic behavior was always interpreted as a result of impurity.
November 24, 2003 (^) Start Presentation
The Forces of Creation I
- Chaos provides nature with a great mechanism for constant innovation.
- We are used to viewing Murphy’s law as something negative: what can go wrong, will go wrong. However, Murphy’s law can also be interpreted as something highly positive: what can grow, eventually will grow.
- Chaos is the great innovator. It brings any and every system constantly to the greatest degree of disorder that it can be in.
- Chaos is built into the very fabric of our universe. At the molecular level, the molecules move around like the balls on the pool table, in total chaos. This is what we measure as entropy. Entropy is being maximized.
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The Forces of Creation II
- Yet, chaos alone would leave us with a universe that is just an accumulation of random white noise. No structure would be retained.
- For structure to be preserved, we also need the opposite force, the great organizer , a force that fosters order, that sifts through the different possibilities, discards the bad ones, and only preserves those that look most promising.
- Three such mechanisms were outlined before. The most powerful among them: Energy is being minimized.
November 24, 2003 (^) Start Presentation
Conclusions
- In the last two lectures, we have looked at predominantly inductive techniques for modeling population dynamics.
- Yet, these techniques have failed to e.g. provide us with a satisfactory model that could help us understand the mechanisms that lead to the oscillatory behavior of the larch bud moth ( zeiraphera diniana ).
- In the next lecture, we shall come up with an improved methodology to deal with these types of systems.
November 24, 2003 (^) Start Presentation
References
- Cellier, F.E. (1991), Continuous System Modeling , Springer-Verlag, New York, Chapter 10.
- Gilpin, M.E. (1979), “Spiral chaos in a predator-prey model,” The American Naturalist , 113 , pp. 306-308.
