# Population Density

- A
- A l
- A x b b b b b
- A12K2 K1 and a21K1 K292c
- A21K1 K2 and K1 a12K292a
- Acknowledgments
- Age and stagestructured linear models relaxing the assumption of population homogeneity
- AN2 hN2 r78b
- Analyze the stability of the logisticlike model you derived in Exercise 51 using the above approach of geometric analysis and linear stability analysis by means of eigenvalues evaluated at the equilibria
- AO cOH O uO
- AS R
- At - 2
- C
- C d1u aPPtll ld1d21214
- Calculate the eigenvalues for the matrices in the table you used for Exercise 22 part 4 and determine the conditions required for each eigenvalue to be less than zero and therefore for a stable equilibrium to exist Then determine the conditions for the ei
- Cc
- CN1 d N1
- Competition between two species mutualism and species invasions
- Contents
- D I0ekd38
- D x xTdxdt dxdtTx xTJ JTx
- D1P1 d2 P2 d3C kL IL dt
- D2 l u2 c2d1 1 u1 c1
- D2 N d2 N d2 NA
- Dfi df2 trJ dN1 dN2
- Di J D J J dNi 1 d N1 dN1
- Diffusion advection the spread of populations and resources and the emergence of spatial patterns
- Discrete logistic growth oscillations and chaos
- DN - 2
- DN dN2 dN dN2 - 2
- DN r
- DN1 dN2 dN1 dN2
- DN1dN2
- DNkdt fkNk N1 Nk1 ak bk
- DNt
- Dt - 2 3
- Dt a2 R1 a R2 R1R2
- Dt K - 2
- Dt K Pmaxh N2
- Dt y k
- Dtdz2
- DX vw
- Dydx eu x du x dx
- Eab Eaeb covA B
- Elambdas eigensysjacob
- F N
- F N dN n 0 dt dt
- F x12x21 dx2
- F2 N m m dt
- Fill in the first two columns of this table you will fill in the last three columns after the next section Matrix trace determinantConditions for stability Conditions for oscillations
- FN N2 D2
- Given that x0 10 at t0 0 and k 05 evaluate the Taylor Series expansion for x t to four terms for t 1 How close is this to simply finding xt by plugging in t 1 and x0 1 on a calculator and using the ex key
- Gt e 1414
- H
- Harvesting and the logistic model
- Homogeneous populations exponential and geometric growth and decay
- How would you alter the differential equations for the resources in Eqs 144 and 145 to reflect N fixation from the atmosphere for one nutrient and uptake from the water for the other nutrient rather than uptake of both resources from the same aqueous pool
- 28
- E R0
- Ii Iii Iv V Vi
- K d x2
- Info - 2 3 4 5
- Inorganic resources mass balance resource uptake and resource use efficiency
- Umax S lsa cd2
- J1 dt 1 K d n
- Jac [evaldiffdn1 n1 evaldiffdn1 n2 evaldiffdn1 nn evaldiffdn2 n1 evaldiffdn2 n2 evaldiffdn2 nn evaldiffdnn n1 evaldiffdnn n2 evaldiffdnn nn nlambdas eigjac
- JV u hP N2
- K - 2
- K 25h
- K a K K a71K N 1N 2911d
- K d 818
- K hPnax Vr2K2 2r2hPmK r2h2P 4rPmKN2 222r
- Nt K
- Xqu
- K0
- K2 N2 a21N1
- KjL M k2LaPMPv aPvPM d1d2
- Kk
- KN akektC
- KN0 akC0 or when
- L sk1
- L U max SUmax
- Lcnt dd1215
- Ln N rt C
- Lsa
- Lw Xw
- Mathematical Ecology of Populations and Ecosystems
- Mathematical toolbox
- MinKP qNi q
- N
- N 0 N K2
- N fN N2 dt
- N K N
- N N0ekt aC0ekt aC0e2kt1423
- N1 3n2
- N2 K2 a21N1 - 2
- Nonlinear models of single populations the continuous time logistic model
- Nr2m2 4KPrm Kph m 2PhK
- P 81pT
- P N
- P q
- P1149
- Perform a bifurcation analysis of the model you derived in Exercise 51 using the above geometric and algebraic techniques
- PiT ekj
- PmaxQ Qmin 11
- Predators and their prey
- Preface
- R 2N dNK
- R P C
- R1K1 a12K2
- RdK r K d Nl rNL2 cK
- RiKi auK2 rK au K2
- Riq2 R1c1R22 R2C22
- RK aiKi K
- RN
- S
- Sa
- Satisfy yourself that N1 0 or the N2 axis is the eigenvector associated with X2 m Therefore the N2 component of a perturbation away from but still near the origin will shrink exponentially at a rate m
- Show that the coexistence equilibrium of R P and C is stable all eigenvalues have negative real parts when I crical see Appendix for Mat Lab commands to do this
- Sketch the vector field plot the nullclines and equilibrium points and determine their stability using both geometric and algebraic methods Compare your results to the Lotka Volterra predatorprey model What do adding interaction terms do to the nullclines
- T
- T ph Af
- Tf Tc Tp
- The minimum viable population is one in which the growth rate is just barely positive Call this population size NlvlaUe From Eq 76 derive an expression in terms of all the other parameters that the actual per capita harvesting rate h must remain below in
- The steady state distribution of forest and prairie
- TrA a
- Transitions between populations and states in landscapes
- U cP - 2
- Vifi0 0
- Prologue
- W sym[flAAf2 fn v sym[n1 n2 nn jacob jacobianwv
- What would a bifurcation diagram for the logistic model look like using K as the control parameter
- What would a Markov matrix look like if no transitions to different classes took place that is if every location stayed in its same class during time step t What is the more general term for this matrix
- X
- 0 i
- 8 1 N N 8Nx Nx8
- D
- Da - 2
- X1 r X2 m
- XB C - 2
- Yo 1 K
- [bo do P SNN52 dt