Optimal Net Energy and Foraging Behavior of the Grus Americana

Optimal Net Energy and Foraging Behavior of the Grus americana Biological Motivation 1. Model berry and crab availability in response to crane foraging • Migrate to ANWR, a rapidly changing wetland 2. Use time dependent prey availabilities to make encounter rates dynamic Goals: develop a predictive model for net energy intake of a whooping crane based upon resource availability in a crane’s territory, and find its maximum Prey Consumed to Net Energy Let i ∈ I = {b, c, ...}, the set containing all prey types that make up the crane’s diet. J= 3. Incorporate integration to account for total time spent in ANWR Final Model: Net Energy Intake Z Tb Z Total time in patch type i sdi , the average encounter rate in patch Probability of successful capture of prey type i Since 95-97% of a crane’s diet consists of blue crabs and Carolina wolfberries, we describe total number of prey consumed in the following way: is then a net energy intake equation where φ describes energy loss. A Switch in Prey Preference Compare the energy efficiency ratios of prey: Eb Ib ` dt 0 , Ec Ic • Ei is the energy gain per prey type i • Ii is the time involved with prey type i which we would expect to be inversely related to the encounter rate: Maximum and Sufficient Let W be the weight of the crane in grams. Then KJ ·W 0.18 · 39 gram 1. Accounts for search and capture within patch is the net energy needed to give the crane a better chance at reproductive success. • Ib = and Eb = 1.3 KJ 1: Find a global maximum net energy intake 2: Compare this energy to the energy deemed necessary • Ic = kc λc and Ec = 139.4 KJ Maximum: Equality of energy efficiency ratios:   kc hc βsnb0 1 t = ln kb hb αsnc0 f − c ln  (mb −`)hb βsnb0 pb −f  ln ,  kJ/crab kJ/berry m2 m2 kJ/day kJ/day kJ/day berries crabs m2 /day days 139.4 1.3 0.065 0.87 878,800 310,900 2,016 3,001.6 3,225.6 6, 837, 064 739,182 0.36 0.01 9,197.928 0.2 20 151 Energy and Switching Point kb λb  α β c f hb hc ` mb mc nb0 nc0 pb pc s kb kc T 2. Accounts for learning curve. So we have that (Tb , Tc ) =  [α(Tc λc pc ) + β(Tb λb pb )] − φ, (αpc λc −mc ) dt− Parameter Estimates for ANWR Energy intake associated with all time being spent in each patch type must be greater than the energy loss associated with spending all time outside of a patch. Tr Z 0 0 Tc λc pc + Tb λb pb . Let α and β be the energy gained by the crane per crab and berry, respectively. Observe that Tc (βpb λb −mb ) dt+ Ti λi pi , i Ti λi pi Conditions for Global Maximum • Overhunting→16 wild birds Energy accumulated while wintering in Texas plays a crucial role in the future survival of the whooping cranes both on an individual and reproductive basis. X Adding Dynamics (mc −`)hc αsnc0 pc −c   Switching Point Conditions and Significance Let our model for net energy intake be called, A. When KJ A ≥ 0.18 · 39 · W, gram This time must be non-negative. Therefore we anticipate: • if 0 < kb hb αsnc0 < kc hc βsnb0 , then f > c • Sufficient energy resources in a crane’s territory • if kb hb αsnc0 > kc hc βsnb0 , then f < c. • A higher probability of reproductive success for the whooping crane. Why is this point important? This gives us incite into the foraging behavior of the crane. The switching point aids in the determination of how time spent hunting and foraging is broken up over the winter. Maximum: (Tb , Tc ) = (4.05, 49.6) → 119, 288.1KJ This satisfies our inequality and suggests 80% of daylight hours are spent hunting and foraging. Switching Point: 3.79 days The initial switch in prey preference from berries to crabs occurs after about 91 hours of foraging. Future Analysis • Consider different search rates of the crane based upon different patch types • Consider the effect of tidal fluctuations on crab availability