Navigation by magnetic signatures in a realistic model of Earth’s magnetic field

Certain animal species use the Earth’s magnetic field (i.e. magnetoreception) alongside their other sensory modalities to navigate long distances that include continents and oceans. It is hypothesized that several animals use geomagnetic parameters, such as field intensity and inclination, to recognize specific locations or regions, potentially enabling migration without a pre-surveyed map. However, it is unknown how animals use geomagnetic information to generate guidance commands, or where in the world this type of strategy would maximize an animal’s fitness. While animal experiments have been invaluable in advancing this area, the phenomenon is difficult to study in vivo or in situ, especially on the global scale where the spatial layout of the geomagnetic field is not constant. Alongside empirical animal experiments, mathematical modeling and simulation are complementary tools that can be used to investigate animal navigation on a global scale, providing insights that can be informative across a number of species. In this study, we present a model in which a simulated animal (i.e. agent) navigates via an algorithm which determines travel heading based on local and goal magnetic signatures (here, combinations of geomagnetic intensity and inclination) in a realistic model of Earth’s magnetic field. By varying parameters of the navigation algorithm, different regions of the world can be made more or less reliable to navigate. We present a mathematical analysis of the system. Our results show that certain regions can be navigated effectively using this strategy when these parameters are properly tuned, while other regions may require more complex navigational strategies. In a real animal, parameters such as these could be tuned by evolution for successful navigation in the animal’s natural range. These results could also help with developing engineered navigation systems that are less reliant on satellite-based methods.


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Although the fact that animals can perform these navigational feats is well established, there is still much unknown about how they accomplish them.
There are many open questions about the neurobiology of sensing magnetic fields: what is the sensor, and how does it transduce information in the nervous system?How do animals use neurally encoded information about the magnetic field to make navigation decisions on long migrations?The answers to these questions almost certainly vary between taxa, and they may even vary within a species for different behaviors.
Progress has been made in answering each of these questions in different species.For instance, at the level of the sensor, evidence in birds (Xu et al 2021) suggests a plausible mechanism in the retina involving cryptochrome 4 for detecting some properties of the magnetic field, such as magnetic inclination, i.e. the angle that magnetic field lines make with the horizontal.Evidence in other taxa (Holland et al 2008, Shaw et al 2015) points toward biogenic magnetic particles that may allow some animals to sense multiple properties of the field, such as inclination and total field intensity.
At the level of behavior, experiments with newly hatched loggerhead sea turtles have shown these animals adjust their swimming direction in response to changing magnetic intensity (Lohmann and Lohmann 1996), as well as to changing magnetic inclination (Lohmann and Lohmann 1994).
Both intensity and inclination vary across the globe (figure 1(a)).In some regions, such as the South Atlantic Ocean, lines of constant intensity and inclination are nearly perpendicular (figure 1(b), light shading), forming a grid in which bicoordinate combinations of intensity and inclination uniquely map to distinct locations on the Earth, at least in the local region.In other regions these lines are nearly parallel (figure 1(b), dark shading, especially near dotted gray lines), which could make bicoordinate navigation more challenging under some algorithmic assumptions (Boström et al 2012), requiring animals to use intensity and inclination as regional rather than localized markers (Lohmann et al 2023), or to use other senses (Lohmann et al 2008) or natural fluid currents (Putman et al 2012(Putman et al , 2015) ) to aid their passage.
Experimental results such as these increase the plausibility of hypotheses relating to how animals may use magnetic information to find their way.In some behaviors of some species, animals may simply use the geomagnetic field as a compass (Wiltschko and Wiltschko 2022), providing information about the animal's heading, which may be paired with an inherited, preferred migratory direction to allow animals to perform vector navigation (Mouritsen andMouritsen 2000, McLaren et al 2022).In other cases, animals may actually use multiple properties of the geomagnetic field, such as intensity and inclination, to locate themselves on a bicoordinate spatial grid, using a so-called map sense (Lohmann et al 2022, Wiltschko and Wiltschko 2022), which may allow animals to determine where they are relative to a goal location.Further, several theoretical and mathematical studies have investigated how animals could use bicoordinate sensory maps, including their potential limitations (Benhamou 1997, 2003, Boström et al 2012, Painter and Plochocka 2019, Hagstrum 2023).
Despite this progress, it remains difficult to test hypotheses about how animals use the Earth's magnetic field in their natural behaviors, outside of a laboratory, when undergoing migrations spanning hundreds or thousands of kilometers.We are particularly interested in understanding whether a map sense based on bicoordinate combinations of geomagnetic intensity and inclination, hereafter called magnetic signatures, is plausible under the varied geomagnetic conditions seen on the Earth today.However, how can one test for such a capability in live animals traveling great distances?In uncontrolled settings, animals will use cues from many senses, such as visual and olfactory cues, to orient and make navigational decisions (e.g.Endres et al 2016).Consequently, it is extremely difficult to determine what role a magnetic sense has in natural behavior at any given instance.Furthermore, some species of interest, including some species of sea turtles, are  ) are plotted.Where a pair of contours intersects in multiple locations (e.g. at locations just west of the Caspian Sea and northeast of Japan), the same magnetic signature is present.Magnetic data originate from the World Magnetic Model, evaluated for the start of 2020.Colors are after Boström et al (2012).(b) Shades of gray indicate the angle between intensity and inclination gradients (equivalently, between their contours).Where the shading is light, the gradients are nearly orthogonal; here, intensity and inclination contours form a grid well suited for bicoordinate navigation.Where the shading is dark, the gradients are nearly parallel; here, intensity and inclination are highly correlated, and magnetic signatures may not be locally unique, although this will not necessarily impede magnetic navigation by animals that use magnetic signatures as markers of general geographic regions (Lohmann et al 2023).The precise locations where the gradients are exactly parallel are indicated by the dotted gray lines (see Boström et al 2012, figure 3).The black cross marks the location of Ascension Island.
endangered and protected, so disturbances to these animals must be limited.
Because of these difficulties, we use computational models to explore what is theoretically possible in animals, and in future bioinspired engineered systems.We turn to models to understand the capabilities and limitations of hypothetical navigation methods.Models are also useful for understanding future, bio-inspired engineering applications of these navigation algorithms in GPS-denied scenarios.
In previous modeling studies (Taylor 2017, 2018, Taylor and Huang 2017, Taylor and Corbin 2019), investigators considered an abstract, configurable bicoordinate magnetic environment in which agents navigated to goal locations characterized by magnetic signatures using a vector formed in magnetic space from the difference between the magnetic field properties of the goal and the local environment.In a subsequent study focused on the North Atlantic, Taylor et al (2021) applied a similar algorithm to a linearized version of the magnetic field properties of that region to better understand how simulated agents may navigate through magnetic environments that more closely resemble those encountered by migratory animals like sea turtles.
In the present study, we consider a similar magnetic navigation algorithm implemented in a realistic global magnetic environment, provided by the World Magnetic Model (WMM) (Chulliat et al 2020).Using a detailed magnetic model based on real-world measurements allows the algorithm to be tested against the particular conditions that real animals face in the wild.Additionally, here we take the analysis of the model further using dynamical systems tools to develop a deeper understanding of the conditions under which the algorithm is likely or unlikely to succeed, and what parametric changes can be made to improve performance.
From our study, we conclude that with appropriate parameter tuning, which might be accomplished by evolution, there are many locations on the Earth where navigation by a magnetic map sense alone is sufficient to reach an intended goal.Although real animals undoubtedly use a combination of sensory modalities to make navigational decisions, and how they use their magnetic sense is still not well understood, we believe our results elevate the plausibility of magnetic cues playing a significant role in long-distance migrations.We emphasize that there is not yet evidence from animal behavior experiments to suggest that animals are using the strategy we describe, or any other particular strategy.Rather, using existing animal observations and data from the literature, our study demonstrates a plausible manner in which animals could use combinations of magnetic properties to successfully navigate from a starting location to a goal point.The results of our study are consistent with previous animal experiments and observations, and can be used to develop novel animal experiment paradigms.The results can also help to develop novel autonomous engineered navigation systems that are less reliant on satellitebased navigation.

Methods
The methods section is organized as follows.First, we list the assumptions present in our model.Next, we present our agent's mathematical motion model, and our implementation of this model.We then present a stability analysis that is used to predict the likelihood of reaching a specified goal point.The methods conclude with a case study that examines migration between Brazil and Ascension Island, a task that green turtles perform (Mortimer andCarr 1987, Mortimer andPortier 1989).
We note here that our stability analysis uses standard approaches that may be found in any text on differential equations or dynamical systems (e.g.Kreyszig and Norminton 1999, Wilson 1999, Blanchard et al 2002, Izhikevich 2010).We present the details of the analysis here for completeness, and for the reader that may not be familiar with these techniques.

Model assumptions
The model makes several simplifying assumptions (see section 4.2 for a discussion of limitations): • Simple kinematics: the agent's position is fully described by its longitude, x, and latitude, y.
Altitude is fixed at sea level.The agent moves freely along the Earth's surface, over both land and sea, in an autonomously chosen direction.• Perfect sensing: the agent continuously senses the total intensity of the magnetic field, F(x, y), and the magnetic inclination (angle of the field relative to horizontal as defined by gravity), I(x, y), at its current location with precision (e.g.no added noise).• Knowledge of goal environment: the agent knows the intensity, F g = F(x g , y g ), and inclination, I g = I(x g , y g ), of the magnetic field at a goal location, (x g , y g ), to which it attempts to navigate.• Zero map knowledge: the agent is never given information on where it or its goal is located (i.e.no coordinates), nor where the agent is relative to its goal (i.e.no heading information), nor does it have access to a global magnetic model or map of any kind (e.g.no 'onboard copy' of the WMM).

• No memory: the agent retains no memory of
where it has been or of prior sensing.Navigation decisions are based on the local and goal magnetic environments.

Model equations
The agent continuously senses its local magnetic environment and updates its velocity.A computed velocity vector, ⃗ d, is calculated as a linear combination of the difference between the goal and local magnetic intensity, ∆F = F g − F(x, y), and the difference between the goal and local magnetic inclination, ∆I = I g − I(x, y): y) . (1) The four entries of the 2-by- are tunable constants that determine the relative importance of intensity (first column) and inclination (second column) on east/west (first row) and north/south (second row) movement.As discussed below, they also determine the efficacy of the algorithm in regions with different magnetic environments (see sections 2.4 and 3).
The agent travels in the direction ⃗ d, up to a maximum speed s.That is, the actual rate of change of the agent's position (i.e.actual velocity) is given by (2) Here, the agent's actual velocity ⃗ f is equal to the computed velocity ⃗ d if the length (magnitude) of ⃗ d is less than or equal to the maximum speed s.Otherwise, if the length of ⃗ d is greater than s, then the actual velocity ⃗ f is in the direction of ⃗ d and the speed is capped at s.

Simulation implementation
To study the navigation algorithm, the model was implemented in MATLAB in a discrete-time simulation, using the forward Euler method for simplicity, with fixed time step ∆t (Gill 2023).Navigation in the simulation is performed iteratively by these steps: 1.The agent senses its local magnetic environment: F(x, y), I(x, y).Magnetic sense data are simulated by querying the WMM (Chulliat et al 2020), with no added noise.2. The agent calculates its velocity, ⃗ f(x, y), according to equation (2). 3. The agent takes a step in the direction of its velocity and in proportion to its speed by updating its longitude and latitude: These steps may repeat for a predetermined number of iterations or until the agent's actual speed, ∥ ⃗ f(x, y)∥, drops below a threshold, which happens when the agent is very near the goal location.If the algorithm would cause the agent's latitude to diverge to |y| > 90 • (i.e.'beyond' the north or south pole), the algorithm is aborted.
For the case study described below in sections 2.5 and 3, the time step was ∆t = 1 (arbitrary time units).The maximum speed was s = 10 −1 degrees per unit time.The navigation algorithm terminated when the actual speed dropped below the threshold ∥ ⃗ f(x, y)∥ < s/100 = 10 −3 degrees per unit time, when 10 4 steps had been taken, or when the agent's latitude diverged to |y| > 90 • , whichever occurred first.

Stability analysis of equilibrium points
Equilibrium points exist where the velocity ⃗ f = ⃗ 0. Assuming the magnetic intensity F(x, y) and magnetic inclination I(x, y) are smooth, locations very near an equilibrium point have a very small velocity.For any positive maximum speed s, there is an arbitrarily small neighborhood around each equilibrium point where ⃗ d(x, y) ⩽ s, and equation ( 2) simplifies to By linearizing ⃗ f around the goal location (x g , y g ), which is always an equilibrium point with ⃗ f(x g , y g ) = ⃗ 0, we obtain a first approximation of the velocity near the goal: (5) The matrix of partial derivatives of the velocity components is the Jacobian matrix of ⃗ f evaluated at the equilibrium point (x g , y g ), or J g , which expands to the following: The stability of the equilibrium point at the goal location (x g , y g ) is determined by the two eigenvalues, λ 1 , λ 2 , of the Jacobian matrix J g .When the real parts of both eigenvalues are negative, the equilibrium point is locally stable; this means that if the agent is sufficiently near its goal (within the 'basin of attraction' of the stable equilibrium point), the agent will certainly move toward it.Otherwise, if one or both of the real parts of the eigenvalues are positive, the equilibrium point is unstable, and the agent will ultimately fail to reach its intended goal.Therefore, in our study, stable points are locations that can be reached (i.e.likely navigation success), while unstable points are locations that cannot be reached (i.e.likely navigation failure).Additionally, trajectories near the equilibrium point have a spiral characteristic if the eigenvalues have imaginary components.
A classic result from linear algebra states that the eigenvalues of a 2-by-2 matrix such as J g are roots of the second-order characteristic polynomial, where tr(J g ) and det(J g ) are the trace and determinant of J g , respectively.Solving for the roots yields and From equation ( 8), it can be shown that the eigenvalues λ 1 , λ 2 of the Jacobian matrix J g have negative real parts-and therefore the equilibrium point at the goal location is locally stable-if and only if tr(J g ) < 0 and det(J g ) > 0. The eigenvalues have imaginary components-and therefore trajectories spiral into or out of the equilibrium point-if and only if tr 2 (J g ) < 4 det(J g ).Thus, the stability of an equilibrium point depends on the navigation parameter matrix,  , and on the magnetic intensity and inclination gradients, ⃗ ∇F and ⃗ ∇I, at the equilibrium point.

Case study
As a case study, we will consider an agent with a goal of reaching Ascension Island in the South Atlantic Ocean, initially positioned west of its goal on the coast of South America.This is similar to a migratory journey made by some populations of green turtles (Mortimer andCarr 1987, Mortimer andPortier 1989).Gradients of magnetic intensity and inclination are relatively orthogonal in this region (see figure 1(b); at Ascension Island, indicated by the black cross, contours are separated by an angle of approximately 60 • ).Based on these regional conditions, we hypothesize that this area is favorable for bicoordinate navigation (Boström et al 2012).The simulation and analysis tools we have presented allow us to quantitatively assess this hypothesis.

Results
Figure 2 shows the results of our case study for six different values of the navigation parameter matrix A. Each plot represents one combination of navigation parameters and shows a trajectory created when the goal location was set to Ascension Island, as well as a heat map that indicates the stability of alternative goal locations around the world.Navigation trajectories plotted in black are based on equation (3) in section 2.3.Heat maps are based on the analysis provided in section 2.4, with greener regions representing goal locations with stable equilibrium points (i.e. points the agent is likely to successfully reach, or navigation success), and yellower areas representing goal locations with unstable equilibrium points (i.e. points the agent is unlikely to reach, or navigation failure).
Figure 2(a) illustrates the case where the navig- The magnetic contours indicate locations on Earth where intensity (blue line) or inclination (red line) are constant and identical to the goal location.Naturally, these lines intersect at the intended goal of Ascension Island (black cross).These contours also intersect at a second location in the South Pacific Ocean, off the western coast of South America.Both contour intersection points therefore have identical magnetic signatures and are equilibrium points for the navigation algorithm.
In figure 2(a), the agent begins its journey from the eastern coast of South America (black circle) and follows the plotted trajectory (thick black line), ultimately reaching Ascension Island.We infer from this result that the equilibrium point at Ascension Island is stable with these parameters.
The background coloring of the map derives from the analysis provided in section 2.4.Keeping A fixed, all locations on the Earth were evaluated as hypothetical goals, and eigenvalues of the Jacobian matrix (equation ( 8)) associated with each location were used to determine the stability of each hypothetical Navigational success can often be obtained with appropriate tuning of parameters.Variations of the parameter matrix A are shown here with trajectories (thick black lines) plotted for an agent starting from the eastern coast of South America (black circles) and seeking the magnetic signature at its goal location of Ascension Island (black crosses).Contours for constant magnetic intensity (blue lines) and inclination (red lines) matching the goal's magnetic signature intersect at the goal and at a second location in the South Pacific Ocean; these two locations therefore have identical magnetic signatures and are both equilibrium points.When parameters are well tuned ((a) and (b)), the goal is reached.When parameters are poorly tuned, the agent may fail to reach any destination, traveling overland in the process ((c) and (d)) or may attempt to navigate to the wrong equilibrium point, also overland ((e) and (f)).Shades of green and yellow indicate the local stability of all potential equilibrium point locations, obtained when the goal is changed to any location in the world, for the given parameters.If an equilibrium point is located in dark or medium green, as are four contour intersections shown here, the equilibrium point is locally stable, and the agent may reach it with the given parameters.If an equilibrium point is located in light green or yellow, as are the other eight contour intersections shown here, the equilibrium point is unstable, and it cannot be reached without modifying the parameters or the navigation algorithm.Regions of stability and instability are often segmented by locations where the intensity and inclination gradients are parallel (dotted gray lines; same as in figure 1(b)).In general, nearly any desired goal location on Earth can be made into a stable equilibrium point with appropriate parameters.equilibrium point.Regions are shaded dark green if hypothetical equilibrium points located within would be stable nodes (negative real eigenvalues), resulting in likely navigation success; in figure 2(a), Ascension Island is located in such a region, which agrees with the example simulation.Yellow shading indicates locations where equilibrium points would be unstable nodes or saddles (real eigenvalues, at least one positive), resulting in likely navigation failure (however, see section 4.2.3); the second magnetic contour intersection point located in the South Pacific Ocean is in such a region, so this equilibrium point is unstable (more precisely, in this case it is a saddle).Medium and light green indicate locations where the equilibrium point is, respectively, a stable spiral sink (complex eigenvalues with negative real parts) or an unstable spiral source (complex eigenvalues with positive real parts).2(e).Changing the parameters does not change the locations of the equilibrium points but does change their stability, converting Ascension Island to a saddle (yellow), and converting the South Pacific equilibrium point to a stable spiral sink (medium green).The spiral characteristic of the latter is clearly seen in the trajectory, which initially approaches Ascension Island before diverting southward along the South American coast and ultimately spiraling into the South Pacific equilibrium point.

Next we consider the case
For an aquatic agent that needs to reach Ascension Island, the first parameter set is clearly superior to the second.Not only does the second parameter set cause the agent to navigate partially overland toward the wrong equilibrium point, but in the case of organisms adapted to warmer waters, such as green turtles, such a southerly trajectory into frigid waters would be lethal.
The other panels of figure 2 illustrate several other parameter variations, including a more efficient navigation attempt (shorter path length; figure 2(b)) and two cases (figures 2(c) and (d)) where no stable equilibrium points exist (contour intersections are in yellow regions) and the agent travels to the poles before the simulation is aborted.
In the process of obtaining our results, we observed that regions of stability and instability tend to be segmented along common lines as the parameters are varied.Frequently, regions of stability and instability are bounded by locations where the magnetic intensity and inclination gradients are parallel (dotted gray lines; these lines in figure 2 are the same as in figure 1(b)).Each parallel-gradient contour acts as a 'fault line' in the stability landscape, i.e. stability cannot be obtained on both sides of the line with any fixed parameter combination (including combinations not shown here).Where a parallelgradient contour runs through an area of the globe, it is not possible for a single set of parameters to ensure that all possible goals in that area (on both sides of the line) are stable equilibrium points.However, an agent undergoing a multi-leg journey through such a region, such as an autonomous underwater vehicle traversing the North Atlantic, could overcome this difficulty by using additional senses to localize its goals or by changing the parameters it uses whenever it changes its goal location from one side of a parallelgradient contour to the other.Furthermore, this difficulty may not apply if agents, rather than seeking a specific goal as they do in this algorithm, use regional magnetic fields as navigational markers or magnetic sign posts to change direction, as appears to be the case for young loggerhead sea turtles (Lohmann et al 2001(Lohmann et al , 2012)), nor would it apply to agents that use entirely different algorithms, such as vector navigation through the use of a magnetic compass and an inherited heading, as many birds likely do (Mouritsen 2018, McLaren et al 2022, 2023).

What the results mean
Many inexperienced migratory animals, such as juvenile loggerhead sea turtles (Lohmann et al 2012) and young songbirds (Mouritsen 2018), manage to navigate alone despite having no prior experience with magnetic fields encountered along their migratory routes.Without assuming the agent has map knowledge of this kind, parameters in our model can generally be adjusted so that the agent succeeds in navigating to the desired goal location.Our results indicate that bicoordinate navigation using magnetic signatures comprising intensity and inclination seems possible in many parts of the world.Furthermore, for the same navigation strategy, while one set of navigation parameters can preclude successful navigation in one part of the world, a different set of parameters can enable successful navigation in the same region.For example, figure 2(a) demonstrates navigation parameters that enable successful navigation from Brazil to Ascension Island, although these parameters would likely not enable successful navigation to the southwestern coast of South America.In contrast, a different set of parameters makes the southwestern coast of South America navigable from Brazil (for agents tolerant of cold waters and capable of overland travel) while making navigation from Brazil to Ascension Island more difficult (e.g.figure 2(f)).It is possible that evolution may have tuned the internal parameters of a given animal (i.e.synaptic strengths, desired goal locations) to enable navigation in the regions of the world that an animal frequents.
An agent could undertake a multi-leg journey across stability 'fault lines' by changing not only the goal magnetic signature on each leg, but also the navigation parameters.Additionally, over evolutionary time, when secular variation of the geomagnetic field is relevant, selective pressure may tune heritable parameters of a navigation algorithm to evolve with changing requirements (Lohmann et al 2001, 2012, 2022, McLaren et al 2023); alternatively, the magnetic environment in which a young animal develops may shape these parameters (Lohmann et al 2022), as appears to occur in sea turtles (Fuxjager et al 2014) and in trout (Putman et al 2014a).

Limitations
While our model and analysis generates encouraging and intriguing results, it is important to point out its limitations and put it into a broader context.

Limitations of the navigation algorithm
In our model, the agent calculates its heading relative to geographic north.We recognize that a magnetically sensitive animal or autonomous robot utilizing a navigation algorithm similar to this one may instead reference magnetic north.For the present work, we found that changing the model to calculate headings relative to magnetic north did not alter the results significantly but complicated the equations and analysis through additional terms related to magnetic declination.We therefore decided to simplify the presentation of this work by computing headings relative to geographic north.
Furthermore, parameterizing speed in terms of degrees per unit time has certain trade-offs.For this work, it eased analysis, implementation, and presentation.However, because lines of longitude are closer together at the poles than at the equator, the speed of an agent traveling at a fixed number of degrees per unit time, when measured in kilometers per unit time, varies depending on latitude and heading.Away from the equator, this biases the agent's heading in our model in the north/south direction, relative to an alternative implementation where speed is measured in kilometers per unit time.In a real-world implementation of this navigation algorithm, in either an animal or robot, the latter parameterization would be more practical.However, our use of degrees rather than kilometers per unit time does not qualitatively change the results of the model or our predictions of its efficacy.

Limitations of the analysis
Although a region in green is categorized as 'locally stable' by this linearization analysis, this says nothing of the size of the basin of attraction for the equilibrium points there.There may be practical limitations in navigating even to goals near the edges of green regions bordering the parallel-gradients contours since there may be duplicate magnetic signatures nearby which are also stable.
Further, if noise is introduced to the magnetic measurements (e.g.representing imprecise sensors), to the magnetic environment itself (e.g.local crustal magnetic anomalies; see Hagstrum 2023, for an important discussion of the practical limitations of mean field models such as the WMM) or to the navigation parameters (e.g.representing individual variation), some equilibrium points that would otherwise be stable might become unstable.

Unimodal vs. multimodal sensation
One limitation of our model is that it only uses magnetoreception to isolate how navigation via the magnetic field might function.In reality, most organisms ranging from bacteria to sea turtles respond to and use a combination of sensory cues including magnetoreception, vision, mechanosensation, audition, and olfaction.Additionally, the salience of these cues is critically dependent on the distance from the target.For example, vision is only useful if an object is within an animal's line of sight and visual acuity.Olfaction is only relevant if odor molecules can be detected and discerned by an animal.The use of multiple sensory modalities in executing a single behavior has been demonstrated in a variety of animals for behaviors that range from locomotion to finding a mate (e.g.Charlton and Cardé 1990, Ekeberg et al 2004, Cardé and Willis 2008).
It is possible that an animal could use a sensory modality such as magnetoreception to navigate at distances that are far away from a goal location (i.e.beyond line of sight and olfaction), while using other sensory modalities such as vision and olfaction to navigate when in the vicinity of a goal location where these cues are more salient and reliable (Lohmann et al 2008).Furthermore, a multimodal navigation strategy may be sufficient to reach goals that our analysis identified as saddle equilibrium points (one positive and one negative real eigenvalue), since saddles attract trajectories in some directions and repel trajectories in others.For example, in figure 2(e), the equilibrium point at Ascension Island (black cross in the yellow, unstable region) is a saddle; the agent first travels toward the island before turning away.If the agent's nearest approach is close enough for other senses, such as olfaction (Endres et al 2016), to detect the goal, that may be enough.For saddle points, the approach is more likely to be close if the negative eigenvalue is larger than the positive eigenvalue, so magnetic navigation parameters might be tuned to control these eigenvalues and bring trajectories closer to saddle points, where other senses may take over for the final leg of the journey.
Other senses would be especially important for animals navigating through regions where lines of constant inclination and intensity are nearly parallel (Boström et al 2012).We found that in these regions it is not possible to select navigation parameters that make goals stable on both sides of the parallelgradient contour.However, it may be possible to tune parameters such that saddle equilibrium points are more common on the unstable side, which could be approached via magnetic navigation and reached under guidance from other modalities.In our future studies, we would like to incorporate additional sensory cues into this model to study how they could be used with this type of navigation strategy.

Influences of other external factors
Another limitation of our model is that it neglects the influence of other external factors that could affect an animal's motion.While our results in the absence of these factors are promising and demonstrate the potential of this type of navigation strategy, they should be considered in future works to better advance our understanding.To provide several examples, for ease of implementation, our model does not include fluid currents such as winds and ocean currents.Fluid currents are of vital importance for aerial and aquatic animals, as these currents could potentially move them off course.Future studies should include these currents to determine their impact on this type of navigation strategy.Previous work by Taylor (2018), Taylor and Corbin (2019) indicate that our signatures-based strategy would likely successfully handle ocean currents.Taylor et al (2021) indicates that for this type of strategy, navigation parameters that yield success may depend on an animal or navigation platform's performance characteristics and external environment (e.g.turning ability, speed, ability to accelerate/decelerate, ocean current strength).For example, Taylor et al (2021) found that for this type of strategy, an animal or platform that has less ability to direct its course (e.g. an animal that spends more of its time passively drifting) might benefit from one set of navigation parameters, whereas an animal or platform with greater control of its course (e.g. an animal that demonstrates strong active swimming) may be more suited to the negative of some of these same parameters.
Geological phenomena such as coastlines and mountain ranges can present barriers that must be surmounted or circumvented in some way, such as pied flycatchers migrating in a way that allows them to avoid phenomena that include the Alps, Mediterranean Sea, and Central Sahara (Lohmann et al 2007).These types of phenomena should be included in future studies.However, previous studies with this type of strategy in a more abstract environment demonstrated how different variants of this strategy could be useful depending on the type of environment.For example, Taylor (2018) demonstrated that if a particular point cues an animal that it must update its motion direction or strategy for the current leg of its journey, then the specific implementation of navigation strategy may not need to be that precise.However, if reaching a particular point is vital for survival, reproduction, or overall migration feasibility, then the strategy implementation may need additional mechanisms to declare navigation success, such as additional sensory modalities.
Finally, our model does not include ecological or dynamical events such as predator evasion or weather-driven disturbances.Although they may take place in a specific location or range of locations, these events have the potential to push an animal severely off course.In the future, we intend to include these events in our model by means of localized intense noise sources.

Other strategies could generate results that are equally compatible with animal data
To put our findings in context, we emphasize that empirical results from animal experiments are compatible with other hypothesized strategies.For example, it is possible that animals do not seek out specific locations ('goals'), but rather, follow a set of rules to stay within an ecologically favorable area.
Our navigation algorithm could easily be modified to do something along these lines.For instance, with our algorithm, an animal could simply try to move towards a small set of points, but not have to reach any one of them (i.e. the tolerance for arriving at one goal before switching to the next in a sequence could be increased).This action could theoretically keep an animal within an ecologically favorable region or help it advance along a migratory route.
Although our navigation algorithm is loosely inspired by experiments with animals, we make extra assumptions with this model that may not hold true in real animals.For instance, the foundational magnetic map work done on juvenile loggerhead sea turtles (for review, see Lohmann et al 2022) shows us that when these animals experience one magnetic field, they respond with a particular swimming direction; in another magnetic field, they may respond with a different swimming direction.Generalizing, swimming direction is a function of the magnetic field.This empirical finding is rather different and less specific than claiming, as we do with this model, that the function is smooth over all values of the magnetic field, and that there is a persistent goal for which the swim velocity would be zero.For instance, empirical animal data might be alternatively explained with a non-smooth piecewise function, where in one broad domain of magnetic field values (e.g.magnetic intensities above a threshold), juvenile turtles respond with one swimming direction, and otherwise they respond with a different swimming direction.That is, their directional response might change discretely rather than smoothly based on a threshold, and there may be no set of magnetic field values corresponding to a 'goal' .This idea is similar to those of 'magnetic waymarks' (Lohmann et al 2007), 'magnetic sign posts' (Wiltschko andWiltschko 2005, McLaren et al 2023), and 'magnetic stop signs' (Wynn et al 2022), where animals sense a magnetic cue, such as an intensity threshold or particular inclination, which triggers an adaptive response, such as a change in heading or stopping travel.
Consistent with this possibility that animals' magnetic response function need not be smooth and continuous, there is evidence that there are multiple, distant locations in the North Atlantic where the magnetic environment elicits no particular swimming direction for juvenile turtles (Putman et al 2015).In these locations, directed swimming may not be important for the animal's survival, as the ocean current will reliably carry them to their next destination, and there is no real danger of veering off course.Drifting with the ocean current is often a good enough strategy for turtles in the North Atlantic Gyre, and occasional directed swimming in crucial areas on the edge of their range may be enough to keep animals on course (Putman et al 2012).
Even if an animal uses a completely different strategy to navigate, the analytical framework we have developed in this study could be of use to analyze that strategy's efficacy.If a strategy can be cast as a dynamical system with a mathematical form of ⃗ x = f (⃗ x,⃗ u), then the same analysis presented in section 2.4 can be performed to determine that strategy's regions of likely success and failure.

Other criteria for success
Real-world constraints on animals and engineered systems will usually require other criteria for successful navigation not considered here.For instance, the amount of energy used by the animal (O'Connell et al 2021), and the amount of time or distance traveled, are critically important to animals' survival.Through the use of model optimization techniques (Kirk 1970), different criteria for success could be achieved.

Future
To better understand how animals, such as sea turtles, might use a magnetoreception-based navigational algorithm such as the one described here in combination with a passive drifting strategy in oceanic environments, future iterations of the model could be implemented in which ocean currents influence the motion of the agent (Putman et al 2012, Painter and Hillen 2015, Painter and Plochocka 2019).In the specific case of loggerhead sea turtles, it is likely that the North Atlantic Gyre has a significant effect on the paths turtles take as they circulate through the region (Lohmann et al 2012).It seems possible that with a magnetic goal location set to a central point, such as the Azores, simulated agents affected by circulating ocean currents may complete a circuit around the North Atlantic, as loggerheads are believed to do.Simultaneously, the magnetoreception-based navigational algorithm may cause the agent to orient in patterns consistent with juvenile loggerheads experiments (Lohmann et al 2012).
Future versions of the model should also consider the real-world limitations of biological and engineered sensors.By limiting the precision and accuracy of magnetic field intensity and inclination sampling, and with the addition of measurement noise, it should be possible to test the limits of the algorithm in more realistic ways (Taylor 2017(Taylor , 2018)).
The model could also be applied to studying the specific migratory patterns of various taxa, such as fish (Putman et al 2014b(Putman et al , 2020) )

Figure 1 .
Figure 1.Magnetic intensity and inclination vary across the globe.(a) Contours for constant magnetic intensity (blue lines; steps of 5 µT) and inclination (red lines; steps of 20• ) are plotted.Where a pair of contours intersects in multiple locations (e.g. at locations just west of the Caspian Sea and northeast of Japan), the same magnetic signature is present.Magnetic data originate from the World Magnetic Model, evaluated for the start of 2020.Colors are afterBoström et al (2012).(b) Shades of gray indicate the angle between intensity and inclination gradients (equivalently, between their contours).Where the shading is light, the gradients are nearly orthogonal; here, intensity and inclination contours form a grid well suited for bicoordinate navigation.Where the shading is dark, the gradients are nearly parallel; here, intensity and inclination are highly correlated, and magnetic signatures may not be locally unique, although this will not necessarily impede magnetic navigation by animals that use magnetic signatures as markers of general geographic regions (Lohmann et al 2023).The precise locations where the gradients are exactly parallel are indicated by the dotted gray lines (see Boström et al 2012, figure3).The black cross marks the location of Ascension Island.

Figure 2 .
Figure2.Navigational success can often be obtained with appropriate tuning of parameters.Variations of the parameter matrix A are shown here with trajectories (thick black lines) plotted for an agent starting from the eastern coast of South America (black circles) and seeking the magnetic signature at its goal location of Ascension Island (black crosses).Contours for constant magnetic intensity (blue lines) and inclination (red lines) matching the goal's magnetic signature intersect at the goal and at a second location in the South Pacific Ocean; these two locations therefore have identical magnetic signatures and are both equilibrium points.When parameters are well tuned ((a) and (b)), the goal is reached.When parameters are poorly tuned, the agent may fail to reach any destination, traveling overland in the process ((c) and (d)) or may attempt to navigate to the wrong equilibrium point, also overland ((e) and (f)).Shades of green and yellow indicate the local stability of all potential equilibrium point locations, obtained when the goal is changed to any location in the world, for the given parameters.If an equilibrium point is located in dark or medium green, as are four contour intersections shown here, the equilibrium point is locally stable, and the agent may reach it with the given parameters.If an equilibrium point is located in light green or yellow, as are the other eight contour intersections shown here, the equilibrium point is unstable, and it cannot be reached without modifying the parameters or the navigation algorithm.Regions of stability and instability are often segmented by locations where the intensity and inclination gradients are parallel (dotted gray lines; same as in figure1(b)).In general, nearly any desired goal location on Earth can be made into a stable equilibrium point with appropriate parameters.
and birds(Kishkinev  et al 2015, Mouritsen et al 2016).Complex, multi-leg journeys could be facilitated in the model by chaining together a sequence of magnetic signature goals, where the agent switches to the next goal once it senses it has arrived at its current goal(Taylor et al 2021,  McLaren et al 2023).Nature 380 59-61 Lohmann K J, Lohmann C M F and Endres C S 2008 The sensory ecology of ocean navigation J. Exp.Biol.211 1719-28 Lohmann K J, Lohmann C M F and Putman N F 2007 Magnetic maps in animals: nature's GPS J. Exp.Biol.210 3697-705 Lohmann K J, Putman N F, Johnsen S and Lohmann C M F 2023 Misconceptions about magnetic maps Zenodo https://doi.org/10.5281/zenodo.10052196Lohmann K J, Putman N F and Lohmann C M F 2012 The magnetic map of hatchling loggerhead sea turtles Curr.Opin.Neurobiol.22 336-42 McLaren J D, Schmaljohann H and Blasius B 2022 Predicting performance of naïve migratory animals, from many wrongs to self-correction Commun.Biol. 5 1-16 McLaren J D, Schmaljohann H and Blasius B 2023 Gauge-and-compass migration: inherited magnetic headings and signposts can adapt to changing geomagnetic landscapes Mov.Ecol.11 37 Mortimer J A and Carr A 1987 Reproduction and migrations of the Ascension Island green turtle (Chelonia mydas) Copeia 1987 103-13 Mortimer J A and Portier K M 1989 Reproductive homing and internesting behavior of the green turtle (Chelonia mydas) at Ascension Island, South Atlantic Ocean Copeia 1989 962-77 Mouritsen H 2018 Long-distance navigation and magnetoreception in migratory animals Nature 558 50-59 Mouritsen H, Heyers D and Güntürkün O 2016 The neural basis of long-distance navigation in birds Annu.Rev. Physiol.78 133-54 Mouritsen H and Mouritsen O 2000 A mathematical expectation model for bird navigation based on the clock-and-compass strategy J. Theor.Biol.207 283-91 National Coordination Office for Space-Based Positioning, Navigation, and Timing 2022 GPS.gov: program funding (available at: www.gps.gov/policy/funding/)O'Connell D, Kehl C E, Taylor B K, Piacenza J, Piacenza S and Faller K J II 2021 A computational framework for studying energetics and resource management in sea turtle migration and autonomous systems J. Theor.Biol.527 110815 Painter K J and Hillen T 2015 Navigating the flow: individual and continuum models for homing in flowing environments J. R. Soc.Interface 12 20150647 Painter K J and Plochocka A Z 2019 Efficiency of island homing by sea turtles under multimodal navigating strategies Ecol.Modelling 391 40-52 Putman N F 2022 Magnetosensation J. Comp.Physiol.A 208 1-7 Putman N F, Meinke A M and Noakes D L G 2014a Rearing in a distorted magnetic field disrupts the 'map sense' of juvenile steelhead trout Biol.Lett. 10 20140169 Putman N F, Scanlan M M, Billman E J, O'Neil J P, Couture R B, Quinn T P, Lohmann K J and Noakes D L G 2014b An inherited magnetic map guides ocean navigation in juvenile Pacific salmon Curr.Biol.24 446-50 Putman N F, Verley P, Endres C S and Lohmann K J 2015 Magnetic navigation behavior and the oceanic ecology of young loggerhead sea turtles J. Exp.Biol.218 1044-50 Putman N F, Verley P, Shay T J and Lohmann K J 2012 Simulating transoceanic migrations of young loggerhead sea turtles: merging magnetic navigation behavior with an ocean circulation model J. Exp.Biol.215 1863-70 Putman N F, Williams C R, Gallagher E P and Dittman A H 2020 A sense of place: pink salmon use a magnetic map for orientation J. Exp.Biol.223 jeb218735 Shaw J, Boyd A, House M, Woodward R, Mathes F, Cowin G, Saunders M and Baer B 2015 Magnetic particle-mediated magnetoreception J. R. Soc.Interface 12 20150499 Storms W, Shockley J and Raquet J 2010 Magnetic field navigation in an indoor environment 2010 Ubiquitous Positioning Indoor Navigation and Location Based Service pp 1-10 (available at: https://ieeexplore.ieee.org/document/5653681)Taylor B K 2017 Bioinspired magnetic reception and multimodal sensing Biol.Cybern.111 287-308 Taylor B K 2018 Bioinspired magnetoreception and navigation using magnetic signatures as waypoints Bioinspir.Biomim.13 046003 Taylor B K, Bernish M K, Pizzuti S A and Kehl C E 2021 A bioinspired navigation strategy that uses magnetic signatures to navigate without GPS in a linearized northern Atlantic Ocean: a simulation study Bioinspir.Biomim.16 046006 Taylor B K and Corbin S 2019 Bioinspired magnetoreception and navigation in nonorthogonal environments using magnetic signatures Bioinspir.Biomim.14 066009 Taylor B K and Huang G 2017 Bioinspired magnetic navigation using magnetic signatures as waypoints Biomimetic and Biohybrid Systems ed M Mangan, M Cutkosky, A Mura, P F Verschure, T Prescott and N Lepora (Springer) pp 48-60