Cost analysis of remotely sensed foraging paths in patchy landscapes with plant anti-herbivore defenses (Patagonia, Argentina)

Abstract

We developed metrics at a landscape scale to evaluate the costs and rewards experienced by large herbivores while foraging in natural vegetation with patchy anti-herbivore plant structures. We show an application of these metrics to the analysis of 16,000 records of positions at successive 1 min intervals of free-ranging ewes (Ovis aries) harnessed with Global-Positioning-System (GPS) loggers, in a large paddock of the Patagonian Monte shrublands (Argentina). Dominant shrubs in the area display numerous anti-herbivore defenses (spiny-resinous leaves, thorny stems, etc.) protecting them from grazing and herbivore trampling. Preferred grasses and forbs constitute a minor part of aboveground plant biomass and grow in relatively open areas among or around shrub patches. We mapped the movement speed of ewes onto high-resolution aerial photographs of the grazed paddocks and estimated costs and rewards along their paths based on algorithms of surface cost theory. Ewes explored areas of sparse vegetation at low speeds compatible with predominant grazing, and increased their speed when crossing denser shrubby patches. The cost algorithm was applied to evaluate daily searching costs as well as grazing rewards in relation to the length of daily searching paths. The observed path lengths and search speeds were consistent with those that compensate costs and rewards of the grazing activities as estimated by the surface cost analysis. We conclude that the technique presented here constitutes a valuable tool to quantify the effect of landscape characteristics on behavioral traits of grazing animals in similar environments.

APPENDIX I

Introduction to the surface cost algorithm

The principle in surface cost analysis is that energy must be expended during travel to do mechanical work, in amounts related to the travel length. Cost values are assigned to points in space according to its distance to a defined target. In a simple case, the surface cost value at a point O (origin, ten distance units from another point T (travel target), is 10. The cost value of a travel starting at T is 0 when T is defined as target, or 10 if O is the target. Costs for all points in space are measured in distance units (du, with length dimensions) respect to a target. Several cost values can be assigned to a single origin, each one with respect to alternative targets.

In most situations, the energy required for transportation is only partially related to the traveled distance, because at some points along the path energy could be obtained or alternatively, extra energy might be required to overcome resistance. Energy can be obtained along the way in the form of back-winds, down-slopes, or energy sources that can be tapped (fuel-tanking places in the case of cars, food in walking animals). Resistance to movement can occur in the form of deep snow, front winds, up-slopes, or any other factor obstructing or deterring movement. In Eastman’s (1989) concept, there is a cost component dependent on the distance traveled, and another component dependent on the relative “friction” encountered along the route. Note that the term “friction” is used in a generalized way and includes both retarding and enhancing effects on the travel energy cost. In order to account for the extra energy expenditures or bonuses occurring at each point in space, costs are expressed in Equivalent Distance Units (eq.du, length dimensions), i.e. the equivalent cost that would result if the travel were performed along a “flat”, no-frictional surface. When costs are computed over digitized images, costs can be conveniently measured in Equivalent Pixels (eq.pixel). Accordingly, costs values at any point in space (c _i ) are expressed as:

$$ \mathbf {c}_{i}(eq.pixel)=(d_i\times f_i) ,$$

(I-1)

where d _i  > 0 is the number of pixels between pixel i to the target pixel, and fi is a so-called frictional, unitless coefficient characterizing extra demands (fi < 0) of energy to travel over pixel i, or fi > 0 if energy gains exist over the same.

Some extensions of the cost algorithm are of interest in landscape ecology applications. In this study, we further define the total cost (${\mathbf{C}}_{n}$) of all possible travels started at any point between two alternative targets (i.e. A, B) as:

$$ {\mathbf{C}}_n(A-B)(eq.pixel)=\Sigma_{i=1\ldots n} {\mathbf{C}}_i $$

(I-2)

where n is the number of pixels between the targets, with average surface cost $({\mathbf{C}}_{d})$:

$$ {\mathbf{C}}_{d}(eq.pixel / pixel) = {\mathbf{C}}_{n}/ n $$

(I-3)

Note that C _d quantifies the average cost value per unit distance along the stretch A-B, independently of the distance between them. Consider the examples on digitized paths displayed in Appendix Table I-1. In a), a subject is assumed to travel along a homogeneous non-frictional area, from any point between two target pixels (A-B) towards either of the extremes, depending on which is closer. Since no energy demand to travel other than the corresponding to mechanical work exists over such area, the c _i value of pixels 1 to 10 strictly depends on the distance to the nearest possible target, which implies that f _i  = −1, for all i. C _n and C _d are computed like in eqs. I–2, I–3 above.

Table I-1 Examples of calculation of the surface cost metrics for travel to targets 10 pìxels from each other. (a) Movement to A or B is over a flat surface, where only the energy expended in mechanical work for transportation is relevant (unit distance/pixel, unit frictional factor/pixel). (b) Movement over pixel 3 at stretch C–D demands three-fold extra energy because of deep snow, while an energy bonus of 0.5 at pixel 4 occurs because of down-slope, resulting in higher C _d than in a). (c) Movement to E or F occurs with energy rewards in the form of backwind, down-slope and food at pixels 3, 4, 7, 9 resulting in C _d   >  0

Let us now consider the case in Table I-1- b) along a stretch C-D where pixel no. 3 might be covered with deep snow that would require a 3-fold energy expenditure respect to a flat surface, and pixel 4 would have a down-slope that would reduce the requirement of mechanical work by 1/2. This is introduced by modifying f ₃ and f ₄ accordingly. Since extra demands of energy occur over the path, the average cost per pixel along C-D is (negative) greater than in case a).

In the case of the stretch E-F, (Table I-1- c) where several pixels along the path are characterized by f _i  > 0 because of down-slopes, food or backwind, the sum of all energy bonuses exceeds the distance cost, and $\user2{C}_{d} > 0$ characterizes a travel along which net energy is obtained.

Consider now a subject seeking to maximize the energy balance at the end of a travel of 10 pixels. In this case, the moving subject would choose the path E-F. Since the cost involved in mechanical work (Appendix Table I-1a) cannot be avoided, the reward function (R) associated with the choice, measured in terms of saved costs, can be defined as:

$$ {\user2 R}(eq.pixel / pixel) = {\user2 C}_{d, (E-F) }- {\user2 C}_{d,( A-B) } ,$$

(I-4)

where the sub-indexes indicate the corresponding stretches in Table I-1.

The cost-reward concepts can be applied to issues of interest in studies on herbivore ecology at a landscape scale. Recent GPS technological developments allow locating the successive positions of GPS-collared animals in space. Consider that A, B, C, D, E,...F in Table I-1 were the successive positions of a collared herbivore, recorded at successive time intervals. Consider overlaying the positions over digitized, remotely sensed landscape images where pixels values would correspond to friction coefficients related to the availability/scarcity of biomass of preferred plants, the thermal field, wind field, geomorphology, distance to water source, etc. or in general any other habitat variable that would be influential in herbivore’s energy balance. Since at any point along the path the animal could decide to continue moving towards the next target or to any other target point in space (including a non-move) the surface cost ${\user2 C}_{d}$ along the whole path A-F would quantify the average value of the instantaneous choices. At the end of a recorded period, the average reward ${\user2 R}$ along the chosen stretches would indicate how efficient the succession of choices was in maximizing the performance with respect to the particular type of friction considered.