Modeling Predator Prey Systems In stable ecosystems we find the populations of predators and prey in a symbiotic balance where the size of one population is regulated by the size of the other In terms of system behaviors this was the example we used for an oscillating system David Tenenbaum GEOG 110 UNC CH Fall 2005 Modeling Predator Prey Systems The Lotka Volterra Model P dP WdP NP dt t dN N N r 1N cPN K dt t Lim t 0 Lim t 0 Note that each rate equation expresses the rate of change in the reservoir at least partially in terms of the reservoir s contents reflecting the exponential component of this system Also N and P appear in both equations reflecting the interdependent nature of the populations in this system giving a hint towards the symmetry and oscillation we expect to see in the resulting behavior David Tenenbaum GEOG 110 UNC CH Fall 2005 Modeling Predator Prey Systems The Lotka Volterra Model Steady state conditions for this system are complex to calculate but can be simplified if we remove the carrying capacity consideration from the prey equation dP 0 P Wd N dt dN dt 0 N r cP These equations are satisfied when either P 0 or N 0 Wd or when r P N c Interestingly the wolf population is a function of deer birth and the deer population a function of wolf death David Tenenbaum GEOG 110 UNC CH Fall 2005 Modeling Predator Prey Systems The Lotka Volterra Model We can understand the system s behavior in terms of the Lotka Volterra phase space diagram Predator isocline A As prey declines so too do predators Predator Population B Once predators get low enough prey begin to recover P r c D A D As predators become abundant prey begin to decline Prey isocline N B Wd C Prey Population C Once there are enough prey predators begin to increase also David Tenenbaum GEOG 110 UNC CH Fall 2005 Modeling of Environmental Systems The next portion of this course will examine the balance flows cycling of three quantities that are present in ecosystems Energy Water Nutrients We will look at each of these at two scales Global Ecosystem Before we can build models of these phenomena we need to have some background on the functioning of these systems with respect to these quantities David Tenenbaum GEOG 110 UNC CH Fall 2005 1 Planck s Law Planck s Law describes the amount of energy technically radiant exitance emitted by a blackbody at a given wavelength at a certain temperature A blackbody is an idealized object that perfectly absorbs all incident electromagnetic radiation and then re radiates it 2 hc M e 5 hc KT 2 1 Where h Planck Constant 6 626E 34 ws2 c speed of light in vacuum 3 0E 8 m s wavelength in meters T temperature in degrees Kelvin K Boltzman constant 1 38054E 23 ws K M blackbody spectral exitance at T Provided you know the temperature of the object you can calculate the amount of energy emitted at a certain wavelength i e for the Sun and the Earth David Tenenbaum GEOG 110 UNC CH Fall 2005 2 Stefan Boltzmann Law Planck s equation provides the spectral exitance for a blackbody at a given temperature and wavelength Integrating Planck s equation over the entire spectrum yields the Stefan Boltzmann equation which gives the total amount of energy emitted by a blackbody at a given temperature M M d 2 hc e 0 Where 2 5 hc KT 1 d T 1 4 Stefan Boltzmann constant 5 676E 8 wm 2K 4 T Temperature in degrees Kelvin Given the temperature we know how much energy a blackbody will emit David Tenenbaum GEOG 110 UNC CH Fall 2005 3 Wien s Displacement Law From Planck s equation we can also derive a law that provides an easy means of finding the wavelength where a blackbody emits the greatest radiation We can do this by taking the first derivative with respect to wavelength and setting it equal to zero to find the wavelength of maximum emission graphically this is equivalent to finding where the slope of the curve is horizontal max dM A d Where 0 max T A 2 798 10 3 mK David Tenenbaum GEOG 110 UNC CH Fall 2005 The Radiation Balance Equation We can describe the net radiation received by the Earth using the Radiation Balance Equation Rn S0 1 0 Ln Where S0 Shortwave radiation from the Sun Albedo describing reflected rad n Ln Net longwave radiation If Rn 0 net gain of energy daytime summer Rn 0 net loss of energy nighttime winter Rn 0 then we have a steady state condition Although this is a very simple equation it explains much of what happens in Earth s ecosystems David Tenenbaum GEOG 110 UNC CH Fall 2005 The Energy Balance Equation We can describe the way the net radiation received by the Earth s surface is partitioned using the Energy Balance Equation Rn LE H Q A Where LE Latent heat H Sensible heat Q Energy stored in the soil A Energy stored in photosynthate How Rn is distributed among the items on the right hand side is determined by the ecosystem biophysical characteristics and has major consequences for ecosystem development and functions David Tenenbaum GEOG 110 UNC CH Fall 2005 Insolation as a F x of Date and Time To calculate the insolation received on the Earth s surface at a given moment I0 h0 Given solar radiance of I0 shining onto the ground at a declination angle of h0 the insolation on the flat ground is I I0 sin h0 To use this simple equation operationally we need to be able to calculate sin h0 as a f x of date and time David Tenenbaum GEOG 110 UNC CH Fall 2005 Insolation as a F x of Date and Time The quantity sin h0 can be calculated using the following equation sin h0 sin sin cos cos cos where is the latitude of the location is the Sun declination is hour angle The hour angle is defined as 0 degrees at local noon takes a negative value before noon and a positive value in the afternoon with 15 degrees per hour away from noon recall that the day is 24 hours long and a full revolution of the Earth is 360 degrees thus 360 degrees 24 hours 15 degrees hour David Tenenbaum GEOG 110 UNC CH Fall 2005 Solar Radiation Received by Bare Ground For bare ground the radiation that the surface receives is determined by the surface reflectance Rr Ri Ra Ri Rr We know that Ri is a function or R0 Sun position However the portion of Ri that is reflected is purely a function of surface physical characteristics The proportion reflected is called reflectance or albedo and we can rewrite the above equation as Ra Ri 1 David Tenenbaum GEOG 110 UNC CH Fall 2005 Radiation Interaction with a Single Leaf To model the movement of energy and water in a terrestrial ecosystem we will begin with the radiation interaction with a