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Systems Ecology

by: Sabina Bosco IV

Systems Ecology NR 575

Sabina Bosco IV
GPA 3.73


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This 41 page Class Notes was uploaded by Sabina Bosco IV on Tuesday September 22, 2015. The Class Notes belongs to NR 575 at Colorado State University taught by Staff in Fall. Since its upload, it has received 44 views. For similar materials see /class/210262/nr-575-colorado-state-university in Natural Sciences at Colorado State University.


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Date Created: 09/22/15
Notes Updated 972006 Formulating Continuous Time Models Formulating Models in Continuous Time There are three simple rules for creating a model Unfortunately nobody knows what they are JW Haefner and W Somerset Maughan Biological Mathematical understanding Abstraction Roadmap39 Developing Dynamic Models A general procedure for formulating models 7 choose state variables 7 diagram relations among state variables 7 formulate balance equations Examples 7 Soil carbon dynamics 7 Disease transmission Useful equations for balance equations The Modeling Process The Real 4 Beliefs 4 Conceptual Model Mathematical 4 Computer O d Model Mo e m each e experimental data observational data dilldiarer SQrQ1 d dt iXS first principles DQ quotQ Q Q 39 other models nm e assumptions educated guesses Balance Equations Changes in state variables over time can be represented as a balance between rates Rate of accumulation of carbon in leaf photosynthetic rate 7 respiration rate 7 translocation rate Rate of population growth birth rate 7 death rate immigration rate 7 emigration rate Rate of change in patch occupancy colonization rate 7 extinction rate Formulating Dynamic Models using Balance Equations Discrete time yzmz F0 y in OW out ow 39 NOTE THE CRITICAL DIFFERENCE HERE Contmuous tlme dt G y in ow rate out ow rate Use these equations to explain why dy discrete time models contain yum Fyt yr jAt in ows and out ows while I continuous time models contain yHAt Fyt z y GyAt in ow rates and out ow rates Discrete time yzzlz F0 y in OW out ow Continuous time 3 Gy in ow rate out ow rate We formulate differential equations as the balance among rates dNdt birth rate death rate immigration rate emigration rate dCdt photosynthetic rate respiration rate dPdt colonization rate extinction rate dNdt mineralization rate immobilization rate dEdt intake rate expenditure rate dIdt infection rate recovery rate mortality rate Break the problem into pieces Focus on each piece Reassemble the pieces into your model Key Point Continuous time dynamic models a system can be divided into discrete compartments or pools that are internally homogeneous and that are connected to other pools by ow of energy materials area or individuals Each compartment is represented by a single state variable h Balance equations describe ows into and out of state variables 0 Model A E S E A PSA7OAEA dB 3 04 35 i 05 Ea The challenge is nding the equations that faithfully portray the rates but that don t have a zillion parameters A general procedure for formulating continuous time dynamic models 0 Choose state variables 0 Diagram relationships among state variables 0 Use diagrams to derive balance equations based on knowledge of rates of change in the state variables Choosing State Variables 0 What are the objectives of your model 0 How much data are available 0 What processes are understood Diagrams for Continuous Time Models rate of additions to Q rate of removals from Q State Variable Q There is no right way to draw model diagrams as long as the diagram helps r c L L you Variables Forester Diagrams How Inlormahon How malenah enevgy In uence stare vanable amen level are equalluns dnvmg variable Forester Diagram Carbon Flow Among Trophic Levels I almospheun I Goa amp 39 respi 39alxon atmospheric cog solid and llqmd wage recipient control donor control Forester Diagram Carbon Flow Among Trophic Levels Exercise Biologically What is wrong with this conceptual model Is there an important ow missing atmospheric lt20i grass v deer W n 9 C A c A a n atmospheric cog sand and mm wasle Forester Diagrams in Power Point 6 atmosphere 5 D 6amp5 6 7 r p 2 Slidezl af73 JD AU AU du mquotmquot quot 39 39 ml m m39 lm 391 l rmnmnllnmmunn ll Examples of Deriving Balance Equations Carbon Budget Model SIR Disease Model Illustrate how a model is derived from simplifying assumptions Don t sweat this part too much Work for understanding It will come at you again Illustrate how to compose balance equations from a box diagram Get this now Purpose Demystify ecological models You could figure this out Get you to focus on the biology symbolized in equations What are units Do they make sense 0 You are not going to be good at this at rst but you will get very good at it Vegetation Soil Exercise Draw in the ows of carbon How would your diagram change if you were modeling nitrogen rather than carbon Herbiv ores Vegetation Soil Herbivores Exercise Add donor or recipient controls that make sense For now have only 1 control for each ow Vegetation i Soil v Herbivores 4 Rates in Continuous Time dV Constant rates E k I aka first order rates Constant relative rates V t a rate constamt Dynamic rates autonomous all 3 VfV Exercise which are dV linear Dynamic rates nonautonomous 7 VfV t t The dynamics of this system are determined by the rate of additions and the rate of losses from each state variable A differential equation summarizes the balance between those rates Given the equation for soil carbon as an example and assuming that all ows are linear ie constant relative rates compose the equations for vegetation and herbivores g le k2V k3S Herbivores Exercise What are paran1eters State variables What are the units of the k s dS Equsz k35 k5V sz kAV k H dH Ek H k1H k7H Herbivores What is wrong with this model How could you add some realism by making rates dynamic g hHhV S m dV hV V hV QH ghH hH hH t Hint per capita herbivore intake rate g Ctime y V Biomass in g C b herbivores g carbon Herbivore consumption of carbon in g C herbivore1 time1 17 Rate of change in herbivore carbon d HHb quot V 7k1H7k7H dz hV dV Ek5VikZVik4ViHb l mV hV mV hV All carbon uxes k1HkZVik3S dz d Vk5V7kZVik4V7H quot V dz hV hV dHa m VjileilgH dz hV Herbivores Exercise Add precipitation as a driving variable controlling carbon uptake Assume there is some small uptake when precipitation 0 and that uptake per gram of carbon in the vegetation increases linearly with increasing precipitation Modify your rate equations to re ect these assumptions r 395 Herbivores Soil Herb ivor es A11 carbon uxes Tk1Hszk3S ail f kmm PVszik4ViH dH 7 mV hV 7D H mV ile IGH dz hV Model of Forest Fuels and Litter I lt 2 3 LIVE TREES Exercise Assume that NPP is a Gompertz function of tree biomass and that all other processes are first order Provide a model of the system mortality Hint D DEAD dB STANDING NPP gt4 fall F FUELS 61 3 pB l lnE quotquot quot decomposition dt in p BO Model of Forest Fuels and Litter quot NPP l pB l ln2 B p 0 D DEAD 73 6D STANDING 6D 6F SIR Model of Disease Transmission Susc eptible Infe cted SIR Model of Disease Transmission and Infected Composing Rate Equations Births Deaths Why are there two rates for the infected individuals Composing Rate Equations Births Deaths Disease Transmission 3 and Infected Composing Rates Disease Transmission by Mass Action aka pseudo mass action Each dot IS a susceptible animal so the total numberofdotsS The are x distribution over space A total area used by the population xed S number of susceptible inA a area traveled by 1 infected animal per unit time aSA number of contacts with susceptible by 1 infected per unit time n proportion of contacts that produce a new infection Let IpA 3S per capita rate of infection th number ofnew infections per infected per unit time BSI number ofnew infections per unit time Composing Rate Equations births deaths transmission bSIR SI ds d1 S dt 31 mdI dR dr dR Composing Rate Equations births deaths transmission recovery gbSIR SI dS d1 S dt mdI I dR CI dR dt Rates in Continuous Time We compose balance dV equations using these Constantrat651 E k building blocks The kinds of blocks we use depend on our Constant relative rates d V kV biOIOgical dt understanding dV Dynamic rates autonomous 7 VfV t d V Dynamic rates nonautonomous E VfV t Constant Rates M gt k M k dt M M0 kt 21 Constant Relative Rates Q kQ dQ E kQ QZQoeikt Understanding relationships between Probabilities of change in state 0 Constant relative rates in continuous time 0 Constant relative rates in discrete time 22 Biological Interpretation of k E dt kQ The best way to understand k is to think ofk as the average time their parameters The average time between events is also the average time a particle remains in a compartment 1 average time between events The relationship between constant relative rates in continuous time and probabilities W Definex as a iandom variable Specifying the duratation ofresidence of a particle in a oomptment Fiist orda kinetics are based on the assumption thatx follows an exponential probability density function P00 ke with mean Thus the avemge retention time is l quot39 k C an The Cumulative distribution function is 1 e e quot which has avalue of 1 M 1 at infinite time By definition it gives theprobabiltythatx r K W quotl 7 a 5 I m By de nition the probability that a particle remains in the compartment for time it l 7 e which means that l The probability that a particle is lost from the compartment at time gt t l e N V The probability that the particle remains in the compartment for time gt t 1 1 ekt 61 Example relationships between constant relative rates and probabilities If a population declines at a constant rate m as a result of mortality then kN Where k is the instanteous rate of mortality dt Nt Noe k N N 6 proportlon of an1mals that were allve at t1me 0 that 0 remain alive at t PS probability of survival from time 0 to time t Ps 67 Pm probablity of mortality from time 0 to time LPquot l e71 24 If we specify t 1 then the probability that a particle moves out of the compartment in a single unit of time is PIe39k and the probability that it remains in the compartment is P e39k Relationship between constant relative rates in discrete and continuous time Recall the fundamental relationship between models in continuous and discrete time dN CN dz dN FNt 2 NI 2 Ni Az dz At 25 Relationship Between Discrete and Continuous Rates dN E Nt Noek NH1 2 Nt detAt At 1 unit of time kd ik NH1 AN 1 lkd kN If we know k What is lambda What is kd 1 N N0e N H1ZNL let t 1 2 N1 Noek 3 zek fromZ N 0 4 hz u from l N ek 1 from 3 and4 lnlk NH1 le Nt det Nlkd solving for kg in terms of 1 kg l l 26 Summarizing relationships between discrete and continuous time rate constants dN k mot kN dt k Nl1739Nl 16 Nz12Nzdez kdek 1 Assume you have know a discrete constant relative rate Ld with units dayS39l What is the discrete rate M with units of hI S39l 27 Dynamic Rates in Continuous Time A modelers toolbox Dynamic rates autonomous all 3 VfV d V Dynamic rates nonautonomous E VfV t 28 A General Approach for Thinking About Dynamic Rates First think of the mass spe cic or per capita etc rate Q E f Q Then think of f Q as a rst order rate modi ed by a feedback term hQ f Q khQ Then mulitply both sides by Q dQ dt khQQ dQ dtQ Dynamic Rates Logistic L9 dQ Q dig a 79 7417 Q dt 29 Homework exercise Sketch the rate equation dNal as a function of N and the state equation N ft to show relative differences for these three cases Explain what is going on particularly with respect to whereyou see the maximum for det dW dtW Dynamic Rates Gompertz dW k W W1771 i it 4quot i Anb ji I W Dynamic Rates Mass action IQ fl f QS E s increasingS or B Q dQ dtQ dt S S S could be another state variable or a driving variable d V rV CVP dt d Pz acVP mVP dt 31 Biological Interpretation of CW aka mass action a total area modeled m2 V number of victims in a a5 area searched per predator per unit time mlpredatortime n proportion of encounters With prey that result in a successful attack unitless Let all 5 0 CV average number of r consumed by each predator per unit I Each dot is prey Assume a random distribution over space 5 er of prey consumed per predator per prey in the population per unit time units predator timequot Dynamic Rates Saturation Increasing Q or 019 anx d QV S 7 Q er m KM S 19 S dtQ EQVMKMS S S S could be another state Variable or a driving Variable Saturation MichaelisMenton dQ V L dtQ quotquot X Km S Vm is maximum rate S is substrate Km is value of S at Vmw2 Vm 100 0 Many applications in enzyme kinetics nutrient uptake functional response Examples of Saturation Functional Response Herbivores H V dB 7 7 CWXB H C B E 7 h B m Cmax maximum consumption rate h B plant biomass h a t constant Predators P dV 7 1V quotquotquotquotquot E 7 1 MW 1 search rate PM hhandling time 1 AhV V prey density Extrinsic Regulation dT k T T dt a Ta is extrinsic quantity another state variable driving variable decision variable Many applications in diffusion Heat flow Newton s Law of Cooling Rate of blood flow between organs is proportional to the difference in blood pressure Dispersal may be proportionate to differences in densities between habitats Combined Feedbacks 0 Combination of positive and negative feeback eg Allee effect photoinhibition plant water relations 0 Example 691 relative rate inhibition facilitation dPPdt mass speci c photosynthetic rate Idriving variable e g light intensity a shape parameter 34 UZAEEuizMiEiEzn Complex Controls I Example plant growth limited by single nutrient N using Monad equation dB N Mm dt k N I BN is biomass production limited by nitrogen g Nis soil nitrogen concentration g m Xis the maximum rate of plant growth when unlimited by N1time kn is the soil N concentration where dBMdtBN Ema2 I What ifother materials eg C and P limit growth as well Approaches to Combining Relative Rates d B B min N C P It max kNN gt kcc gt kPP Law ofm1n1mum dB M N L L It max k N kc C M 13 Multiplicative dim N L L dz 3 kN N kc C kP P Average 1 1 71 1 3 B m l L L P Res1stance dt 3 kN N kc C kP P Which of these works best depends on biology of system Three more things 0 Checking units in continuous time models 0 Implementing continuous models using discrete time approximations Algebra of box models 36 httpWWWphysicsuoguelphcatutorialsdimanaly ma iVIENSIONAL ANALYSIS mammmm quotmnmm Rules for Units Dimensions of most ecological problems length mass time Dimensions determine the units Counts of things are dimensionless and hence unitless The units ofthe left hand side ofdynamic ecological models always have time in the denominator The numerator of the left hand side usually have mass number or ength or some combination ie mass lengch in the numerator The units of the right hand side must cancel to the units of the left hand side Any quantities that are summed or subtracted must have the same units dV 2V 1371th The dimensions of of prey and predator density ie V P are length39z What are the units of it and h 7 Use minutes for the time dimension and meters for the length dimension dV 2V dtP 1 th dm39Z lm39z dtm39z 1 lhm39z There fore we know a couple of things 1 The unit ofthe lhs must t39l min391 2hm39Z must be unitless Z Therefore the units of h min min n 72 ml L dtm 2 mi min 38 Solving continuous time dynamic models with multiple state variables using discrete approximations compute all ows before you update state variables PredatorPrey Example This way Not this way Start S Kn rV tart V V rV VP Vqu VP P P a VP mP Pm a VP End PM mp V VVm VM P PPm PM Exercise Why End Algebra at Steady State Whenever the ows into a compartment can be assumed to equal the ows out the material in the compartment is at steady state When this occurs amount 1n compartment m Fbut T res1dence t1me D M gt Fm F0141 39 Exercises The standing stock carbon contained in live terrestrial biomass is approximately 56 X 1017 g Net primary production of the earths terrestrial ecosystem is 5 X 1016 g C yr The standing stock carbon contained in live marine biomass is approximately 2 X 1015 g Net primary production of marine ecosystems is 25 X1016g Cyr What is the average residence time of carbon in terrestrial and marine ecosystems Using What you learned about rst order kinetics and mean retention time to s ow M F F 7 M 55 X IDquotgC Fm 39 5 K mvsmCW Fm Fm T g n 2y remasmar mamass M quot 2 gtlt m SgAc Fm 39 25 x VD QACyv T 0081 Mama biomass FF kAI m out 1 From rst oder kinetics T E kzi T Substituting for k EnzFbuIZ T Roadmap Developing Dvnamic Models The general equation for dynamic models Translating the general equation to speci c models using balance relations A general procedure for formulating models 7 choose state variables 7 diagram relations among state variables 7 formulate balance equations Examples 7 Soil carbon dynamics 7 Disease transmission Useful equations for balance equations Miscellaneous useful knowledge about rates 41


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